Abstract
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor \(\beta >0\) per edge. It arises as the \(q\to 0\) limit of the \(q\)-state random cluster model with \(p=\beta q\). We prove that in dimensions \(d\geqslant 3\) the arboreal gas undergoes a percolation phase transition. This contrasts with the case of \(d=2\) where no percolation transition occurs.
The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane \(\mathbb{H}^{0|2}\). This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the \(\mathbb{H}^{0|2}\) model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.
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1 Introduction
This paper has two distinct motivations. The first is to study the percolative properties of the arboreal gas, and the second is to understand spontaneously broken continuous symmetries. We first present our results from the percolation perspective, and then turn to continuous symmetries.
1.1 Main results for the arboreal gas
The arboreal gas is the uniform measure on (unrooted spanning) forests of a weighted graph. More precisely, given an undirected graph \(G=(\Lambda ,E)\), a forest \(F=(\Lambda ,E(F))\) is an acyclic subgraph of \(G\) having the same vertex set as \(G\). Given an edge weight \(\beta >0\) (inverse temperature) and a vertex weight \(h\geqslant 0\) (external field), the probability of a forest \(F\) under the arboreal gas measure is
where \(T\in F\) denotes that \(T\) is a tree in the forest, i.e., a connected component of \(F\), \(|E(F)|\) is the number of edges in \(F\), and \(|V(T)|\) is the number of vertices in \(T\). The arboreal gas is also known as the (weighted) uniform forest model, as Bernoulli bond percolation conditioned to be acyclic, and as the \(q\to 0\) limit of the \(q\)-state random cluster model with \(p/q\) converging to \(\beta \), see [57].
We study the arboreal gas on a sequence of tori \(\Lambda _{N} = \mathbb{Z}^{d}/L^{N}\mathbb{Z}^{d}\) with \(L\) fixed and \(N\to \infty \). To simplify notation, we will use \(\Lambda _{N}\) to denote both the graph and its vertex set. From the percolation point of view, the most fundamental question concerns whether a typical forest \(F\) under the law (1.1) contains a giant tree. In all dimensions, elementary arguments show that giant trees can exist only if \(h=0\) and if \(\beta \) is large enough, in the sense that connection probabilities decay exponentially whenever \(h>0\) or \(\beta \) is small; see Appendix A.2.
The existence of a percolative phase for \(h=0\) and \(\beta \) large does not, however, follow from standard techniques. The subtlety of the existence of a percolative phase is perhaps best evidenced by considering the case \(d=2\): in this case giant trees do not exist for any \(\beta >0\) [20]. Our main result is that for \(d\geqslant 3\) giant trees do exist for \(\beta \) large and \(h=0\), and that truncated correlations have massless free field decay. To state our result precisely, let \(\{0 \leftrightarrow x\}\) denote the event that 0 and \(x\) are connected, i.e., in the same tree.
Theorem 1.1
Let \(d\geqslant 3\) and \(L \geqslant L_{0}(d)\). Then there is \(\beta _{0} \in (0,\infty )\) such that for \(\beta \geqslant \beta _{0}\) there exist \(\zeta _{d}(\beta ) = 1-O(1/\beta )\), \({\mathsf{c}}(\beta ) = {\mathsf{c}}+ O(1/\beta )\) with \({\mathsf{c}}>0\), and \(\kappa >0\) such that
where \(|x|\) denotes the Euclidean norm.
Numerical evidence for this phase transition of the arboreal gas was given in [40]. More broadly our work was inspired by [21, 37, 38, 40, 60, 61], and we discuss further motivation later.
Although both the arboreal gas and Bernoulli bond percolation have phase transitions for \(d\geqslant 3\), the supercritical phases of these models behave very differently: (1.2) shows that the arboreal gas behaves like a critical model even in the supercritical phase, in the sense that it has massless free field truncated correlation decay. While this behaviour looks unusual when viewed through the lens of supercritical percolation, it is natural from the viewpoint of broken continuous symmetries. We will return to this point in Sect. 1.2.
Theorem 1.1 concerns the arboreal gas on large finite tori in zero external field (i.e., \(h=0\)). Another limit to consider the arboreal gas in is the weak infinite volume limit. To this end, we consider the limit obtained by first taking \(N\to \infty \) with \(h>0\) and then taking \(h\downarrow 0\). In a manner similar to that for Bernoulli bond percolation in [47, Sect. 5] and [2, Sect. 2.2], the external field is equivalent to considering the arboreal gas on an extended graph \(G^{\mathfrak{g}} = (\Lambda \cup \{\mathfrak{g}\}, E\cup E^{ \mathfrak{g}})\) where \(E^{\mathfrak{g}} = \Lambda \times \{\mathfrak{g}\}\) and each edge in \(E^{\mathfrak{g}}\) has weight \(h\). The additional vertex \(\mathfrak{g}\) is called the ghost vertex. The measure (1.1) is then obtained by forgetting the connections to the ghost. This rephrases that the product in (1.1) is equivalent to connecting a uniformly chosen vertex in each tree \(T\) to \(\mathfrak{g}\) with probability \(h|V(T)|/(1+h|V(T)|)\). For vertices \(x,y \in \Lambda \), we continue to denote by \(\{x\leftrightarrow y\}\) the event that \(x\) and \(y\) are connected in the random forest subgraph of \(G\) with law (1.1), i.e., \(\{x\leftrightarrow y\}\) denotes the event that there is a path from \(x\) to \(y\) in the random subgraph, and that this (necessarily unique) path does not contain \(\mathfrak{g}\). We write \(\{x\leftrightarrow \mathfrak{g}\}\) to denote the event that \(x\) is connected to \(\mathfrak{g}\).
The event \(\{0\leftrightarrow \mathfrak{g}\}\) is a finite volume proxy for the event that the tree \(T_{0}\) containing 0 becomes infinite in the infinite volume limit when \(h\downarrow 0\). Indeed, let us define
and let \(\mathbb{P}_{\beta}^{\mathbb{Z}^{d}}\) be any (possibly subsequential) weak infinite volume limit \(\lim _{h\downarrow 0}\lim _{N\to \infty}\mathbb{P}^{\Lambda _{N}}_{ \beta ,h}\). Then
see Proposition A.6. By a stochastic domination argument it is straightforward to show that
where \(p_{c}(d)\) is the critical probability for Bernoulli bond percolation on \(\mathbb{Z}^{d}\), see Proposition A.3. When \(d=2\), \(\theta _{2}(\beta ) = 0\) for all \(\beta >0\) by [20, Sect. 4.2]. The next theorem shows that for \(d\geqslant 3\) the arboreal gas also has a phase transition in this infinite volume limit.
Theorem 1.2
Let \(d\geqslant 3\) and \(L \geqslant L_{0}(d)\). Then there is \(\beta _{0} \in (0,\infty )\) such that for \(\beta \geqslant \beta _{0}\) the limit (1.3) exists and
where \(\zeta _{d}(\beta )\) is the finite volume density of the tree containing 0 from Theorem 1.1.
In fact, our proof shows that \(\theta _{d}(\beta )\sim 1-c/\beta \) with \(c=(-\Delta ^{\mathbb{Z}^{d}})^{-1}(0,0)>0\) the expected time a simple random walk spends at the origin. This behaviour is different from that of Bernoulli bond percolation and more generally that of the random cluster model with \(q>0\). For these models the percolation probability is governed by Peierls’ contours and is \(1-O((1-p)^{2d})\) by [70, Remark 5.10].
That the arboreal gas behaves critically within its supercritical phase can be further quantified in terms of the following truncated two-point functions:
Theorem 1.3
Under the assumptions of Theorem 1.2, for \(\beta \geqslant \beta _{0}\), the limits (1.7)–(1.8) exist and there exist constants \({\mathsf{c}}_{i}(\beta ) = {\mathsf{c}}_{i}+O(1/\beta )\) and \(\kappa >0\) such that
The constants satisfy \(({\mathsf{c}}_{2}(\beta )/{\mathsf{c}}_{1}(\beta )^{2})\theta _{d}(\beta )^{2}=1\) and \({\mathsf{c}}(\beta ) = 2{\mathsf{c}}_{1}(\beta )\), \({\mathsf{c}}(\beta )\) from Theorem 1.1.
Further results could be deduced from our analysis, but to maintain focus we have not carried these out in detail. We mention some of them below in Sect. 1.4 when discussing our results and open problems.
1.2 The \(\mathbb{H}^{0|2}\) model and its continuous symmetries
In [37, 38], the arboreal gas was related to a fermionic field theory and a supersymmetric non-linear sigma model with target space one half of the degenerate super-sphere \(\mathbb{S}^{0|2}\). In [20] this was reinterpreted as a non-linear sigma model with hyperbolic target space \(\mathbb{H}^{0|2}\), which we refer to as the \(\mathbb{H}^{0|2}\) model for short. The reinterpretation was essential in [20]; it is less essential for the present work, but nevertheless we continue to use the \(\mathbb{H}^{0|2}\) formulation of the model.
Briefly, the \(\mathbb{H}^{0|2}\) model is defined as follows, see [20, Sect. 2] for further details. For every vertex \(x\in \Lambda \), there are two (anticommuting) Grassmann variables \(\xi _{x}\) and \(\eta _{x}\) and we then set
Thus the \(z_{x}\) commute with each other and with the odd elements \(\xi _{x}\) and \(\eta _{x}\). The formal triples \(u_{x}=(\xi _{x},\eta _{x},z_{x})\) are supervectors with two odd components \(\xi _{x}\), \(\eta _{x}\) and an even component \(z_{x}\). These supervectors satisfy the sigma model constraint \(u_{x}\cdot u_{x}=-1\) for the super inner product
In analogy with the tetrahedral representation of the \(q\)-state Potts model, see [26, Sect. 2.2], the sigma model constraint can be thought of as \(u_{x}\cdot u_{x} = q-1\) with \(q=0\). The constraint is also reminiscent of the embedding of the hyperbolic space \(\mathbb{H}^{2}\) in \(\mathbb{R}^{3}\) equipped with the standard quadratic form with Lorentzian signature \((1, 1, -1)\). Indeed, \(-\xi _{x}\eta _{y}-\xi _{y}\eta _{x}\) is the fermionic analogue of the Euclidean inner product on \(\mathbb{R}^{2}\).
Let \(F\) be a (non-commutative) polynomial in the variables \(\{\xi _{x},\eta _{x}\}_{x\in \Lambda}\). The expectation of \(F\) with respect to the \(\mathbb{H}^{0|2}\) model is
In this expression, \(\int \prod _{x\in \Lambda} \partial _{\eta _{x}}\partial _{\xi _{x}}\) denotes the Grassmann integral (i.e., the coefficient of the top degree monomial of the integrand), \(Z_{\beta ,h}\) is a normalising constant, and
where \(xy\in E(\Lambda )\) denotes that \(x\) and \(y\) are nearest neighbours (counting every pair once), and the inner products are given by (1.12). The factors \(1/z_{x}\) in (1.13) are the canonical fermionic volume form invariant under the symmetries associated with (1.12) as discussed further below.
As explained in [20, Sect. 2.1] (see also [37] where such relations were first observed) connection and edge probabilities of the arboreal gas are equivalent to correlation functions of the \(\mathbb{H}^{0|2}\) model. The following proposition summarises the relations we need, see Appendix A for the proof.
Proposition 1.4
For any finite graph \(G\), any \(\beta \geqslant 0\) and \(h \geqslant 0\),
and the normalising constants in (1.1) and (1.13) are equal. In particular,
These relations resemble those between the Potts model and the random cluster model, giving further credence to our proposal that the \(\mathbb{H}^{0|2}\) model may be interpreted as the 0-state Potts model, with the arboreal gas playing the role of the 0-state random cluster model. Nevertheless, there are important differences from the \(q\)-state Potts model with \(q \geqslant 2\). Chief amongst them is that the \(\mathbb{H}^{0|2}\) model has continuous symmetries. To make this precise, let
One way to understand the significance of \(T\), \(\bar{T}\) is via the identities \(\langle TF \rangle _{\beta ,0}=\langle \bar{T} F \rangle _{\beta ,0}=0\) for any polynomial \(F\) in the variables \(\xi \) and \(\eta \); see [20, Sect. 2.2]. For example, \(\langle T\xi _{0} \rangle _{\beta ,0}=\langle z_{0} \rangle _{\beta ,0}=0\). Identities derived in this way are conventionally called Ward identities.
The maps \(T\) and \(\bar{T}\) are infinitesimal generators of two global internal supersymmetries of the \(\mathbb{H}^{0|2}\) model. These supersymmetries are explicitly broken if \(h \neq 0\). They are analogues of infinitesimal Lorentz boosts or infinitesimal rotations. Together with a further internal symmetry corresponding to rotations in the \(\xi \), \(\eta \) plane, these operators generate the symmetry algebra \(\mathfrak{osp}(1|2)\) of the \(\mathbb{H}^{0|2}\) model. For details and further explanations we again refer to [20, Sect. 2.2]. As generators of continuous symmetries, \(T\) and \(\bar{T}\) imply Ward identities that are not available for the Potts model with \(q\geqslant 2\). These identities are crucial for our analysis and will be discussed below.
The phase transition of the arboreal gas corresponds to a spontaneous breaking of the above supersymmetries in the infinite volume limit. By spontaneous symmetry breaking we mean that there is an observable \(F\) for which \(\lim _{N\to \infty}\lim _{h\downarrow 0}\langle F \rangle _{\beta ,h} \neq \lim _{h\downarrow 0}\lim _{N\to \infty}\langle F \rangle _{ \beta ,h}\). Indeed, this is shown in our next theorem for the \(\mathbb{H}^{0|2}\) model from which Theorems 1.2 and 1.3 follow immediately by (1.15)–(1.17) (except for the same statements relating the constants, which we omitted here). A similar reformulation applies to Theorem 1.1.
Theorem 1.5
Let \(d\geqslant 3\) and \(L\geqslant L_{0}(d)\). There exists \(\beta _{0}\in (0,\infty )\) and constants \(\theta _{d}(\beta )=1+O(1/\beta )\) and \({\mathsf{c}}_{i}(\beta ) = {\mathsf{c}}_{i}+O(1/\beta )\) and \(\kappa >0\) (all dependent on \(d\)) such that for \(\beta \geqslant \beta _{0}\),
In particular,
In fact, the constants \({\mathsf{c}}_{i}(\beta )\) both satisfy \({\mathsf{c}}_{i}(\beta ) = (c_{d})^{i} + O(1/\beta )\), where \(c_{d}\) is the leading constant in the asymptotics of the Green function of the Laplacian \(-\Delta ^{\mathbb{Z}^{d}}\) on \(\mathbb{Z}^{d}\):
Our proof of Theorem 1.5 is by a rigorous renormalisation group analysis aided by Ward identities. We set \(\psi =\sqrt{\beta}\eta \) and \(\bar{\psi}= \sqrt{\beta}\xi \). Algebra then shows the fermionic density in (1.13) is equivalent to
where the 1 in the quadratic term arises from putting the \({\mathbb{H} } ^{0|2}\) volume form (see (1.13)) into the exponential, i.e.,
and \(\mathcal{E}_{d}=\{e_{1},\dots , e_{2d}\}\) are the standard unit vectors (where \(e_{d+j}=-e_{j}\)). The reformulation (1.25) looks very much like a fermionic version of the \(\varphi ^{4}\) spin model. However, the following differences are important:
(1) Due to the fermionic nature of the field, and because the fermionic field only has two components (different for example from the case of Dirac fermions with four components), the quartic term actually has gradients in it: denoting the discrete gradient in direction \(e \in \mathcal {E} _{d}\) by \((\nabla _{e}\psi )_{x} = \psi _{x+e}-\psi _{x}\), the quartic term can be written as
where we introduced the shorthand notation
(2) The coupling constants \(\frac{1}{\beta}(1+h)\) of the quadratic and \(\frac{1}{\beta}\) of the quartic terms are related, and they are equal in the case \(h=0\) of ultimate interest. This relation is due to the geometric origin of the model as a non-linear sigma model and analogous relations are present in intrinsic coordinates for other sigma models like the vector \(O(n)\) model. We remark that if the coupling constant of the quartic term was much smaller than that of the quadratic term (so \(h\gg 0\)) the study of the model would reduce to an exercise in fermionic cluster expansions (but see Appendix A.2 for even simpler arguments in this case).
To study the case of equal coupling constants, we will first consider their renormalisation group trajectories as a one parameter family among the set of all renormalisation group trajectories obtained by allowing the initial quadratic and quartic couplings to vary independent of one another. We will then place the equal-initial-coupling trajectories on the critical manifold of the renormalisation group dynamical system using the following Ward identity for the \(\mathbb{H}^{0|2}\) model:
where \(T\) is the symmetry generator (1.19); in the second equality we have used [20, Lemma 2.3].
After taking into account the two points above (in particular that the flow of the expanding quadratic term is constrained by the Ward identity), power counting heuristics predict that the lower critical dimension for spontaneous symmetry breaking with free field low temperature fluctuations is two for the \(\mathbb{H}^{0|2}\) model. We expect that these considerations generalise to all non-linear sigma models with continuous symmetry, in agreement with the Goldstone mechanism. In conjunction with [20], our results rigorously establish that the lower critical dimension is two for the \(\mathbb{H}^{0|2}\) model.
1.3 Background on non-linear sigma models and renormalisation
The low temperature renormalisation group analysis of non-linear sigma models with non-abelian continuous symmetry is a notorious problem that was famously considered by Balaban for the case of \(O(n)\) symmetry, see [10, 11] and references therein. Our comparatively simple analysis of the \(\mathbb{H}^{0|2}\) model, which is a non-linear sigma model with non-abelian continuous \(OSp(1|2)\) symmetry, is made possible mainly by the fact that it does not suffer a large field problem because it has a fermionic representation. Our approach to the \(\mathbb{H}^{0|2}\) model differs from Balaban’s approach to the \(O(n)\) model on a conceptual level, in that it is based on intrinsic coordinates as opposed to extrinsic ones. In the extrinsic approach of Balaban the sphere \({\mathbb{S} } ^{n-1}\) is embedded into \(\mathbb{R}^{n}\) and the renormalised action evolves as a function of \(\mathbb{R}^{n}\)-valued fields, manifestly preserving \(O(n)\) symmetry. The intrinsic approach we use, which is similar to the one taken in the physics literature (see [79]), is based on local coordinates for the target space \({\mathbb{H} } ^{0|2}\). The renormalised action does not have the \(OSp(1|2)\) symmetry of the model, and Ward identities are used a posteriori to enforce the essential constraints (relations between couplings) due to the symmetry. It is unclear to us how to implement an extrinsic approach in our situation of \(OSp(1|2)\) symmetry, and more generally for noncompact symmetries.
Somewhat remarkably, despite its simplicity, the \(\mathbb{H}^{0|2}\) model has all of the main features present in the non-abelian \(O(n)\) models, including: absence of spontaneous symmetry breaking in 2d (proven in [20]); mass generation in 2d (conjectured in [38]); and a spontaneous symmetry breaking phase transition with massless low temperature fluctuations in \(d\geqslant 3\) (the main result of this work).
The \(\mathbb{H}^{0|2}\) model is a member of the family of hyperbolic sigma models with target spaces \(\mathbb{H}^{n|2m}\), see [39] for a discussion of some aspects of this. By supersymmetric localisation the observables of the \(\mathbb{H}^{0|2}\) model considered in Theorem 1.5 are equivalent to the analogous ones of the non-linear sigma model with target \(\mathbb{H}^{2|4}\). While this relation does not play a role in this paper, it leads to a more direct representation of the continuous symmetry breaking observed here. In brief, in the \(\mathbb{H}^{2|4}\) model each vertex comes equipped with two real and four Grassmann fields. By expressing these fields in horospherical coordinates one of the real fields and the four Grassmann fields can be integrated out. The marginal distribution of the remaining real field, which is called the \(t\)-field, may be viewed as a ‘\(\nabla \phi \)’ random surface model, albeit with a nonconvex and nonlocal Hamiltonian. By this we mean that the potential is invariant under the global translation \(t_{x}\mapsto t_{x} +r\) for \(r\in \mathbb{R}\). See [20] for more details, where this perspective was used to prove the absence of symmetry breaking in \(d=2\). The full \(\mathbb{H}^{n|2m}\) family has been important for advancing our understanding of other aspects of these models [20, 39]. Of particular note, we mention that the \(\mathbb{H}^{2|2}\) model has received substantial prior attention due to its exact connection to linearly reinforced random walks and its motivation from random matrix theory, see [42, 72, 78, 80, 81].
For hyperbolic sigma models with target \(\mathbb{H}^{n}\), \(n \geqslant 1\), spontaneous symmetry breaking for all \(\beta >0\) was shown in [78], and with target \(\mathbb{H}^{2|2}\) for \(\beta \) large in [42] (see also [43]). For motivation from random matrix theory and the Anderson transition see [76, 77]. These proofs make essential use of the horospherical coordinates mentioned above. Moreover, the proof of symmetry breaking for the \(\mathbb{H}^{2|2}\) model in [42] relies on an infinite number of Ward identifies resulting from supersymmetric localisation. These identities are absent in the \(\mathbb{H}^{0|2}\) model, limiting the applicability of the methods of [42] to our setting. At the same time, the \(\mathbb{H}^{2|2}\) model has no purely fermionic representation, and so our methods do not apply there, at least without significant further developments.
Introductions to fermionic renormalisation include [22, 66, 73], see also [53]. Recent probabilistic applications of these approaches to fermionic renormalisation include the study of interacting dimers [51, 52] and two-dimensional finite range Ising models [7, 8, 49, 50]. Our organisation of the renormalisation group is instead based on a finite range decomposition and polymer coordinates, and follows [28] and its further developments in [12, 16, 17, 29–32, 36]. This approach has its origins in [33]. For an introduction to this approach in a hierarchical bosonic context see [18]. Previous applications of this approach include the study of 4d weakly self-avoiding walks [14, 15]; the nearest-neighbour critical 4d \(|\varphi |^{4}\) model [13, 75] and long-range versions thereof [63, 74]; the ultraviolet \(\varphi ^{4}_{3}\) problem [34, 35]; analysis of the Kosterlitz–Thousless transition of the 2d Coulomb gas [41, 45]; the Cauchy–Born problem [1]; and others.
While the construction of the bulk renormalisation group flow is simpler for the intrinsic representation of the \(\mathbb{H}^{0|2}\) model than in many of the previous references, a crucial novelty of our present work is the combination of the finite range renormalisation group approach with Ward identities, together with a precise analysis of a nontrivial zero mode. This has enabled us to apply these methods to a non-linear sigma model in the phase of broken symmetry. It would be extremely interesting to understand this approach for bosonic non-linear sigma models where, while ‘large fields’ cause serious complications, the formal perturbative analysis is very much in parallel to the fermionic version we study in this paper. Ward identities of a different type have previously been used in the renormalisation group analyses in [9] and [23] and many follow-up works including [51, 52]. Finally, we mention that Theorem 1.1 yields quantitative finite volume statements. The proof implements a rigorous finite size analysis along the lines of that proposed in [27]. It would be very interesting to extend this to even higher precision as discussed in Sect. 1.4 below.
1.4 Future directions for the arboreal gas
In this section we discuss several interesting open directions, including the geometric structure of the weak infinite volume limits of the arboreal gas and its relation to the uniform spanning tree, and a conjectural finite size universality similar to Wigner–Dyson universality from random matrix theory.
1.4.1 Finite volume behaviour
The detailed finite volume behaviour of the arboreal gas would be very interesting to understand beyond the precision of Theorem 1.1. On the complete graph at supercritical temperatures it is known that there is a unique macroscopic cluster, and that there are an unbounded number of clusters whose sizes are of order \(|\Lambda |^{2/3}\) [64]. The fluctuations of the macroscopic cluster are non-Gaussian of scale \(|\Lambda |^{2/3}\) and the distribution of the ordered cluster sizes of the mesoscopic clusters has been determined [64]. The joint law of the mesoscopic clusters can be characterised [65, Sect. 1.4.3]. Intriguingly, \(|\Lambda |^{2/3}\) is the size of the largest tree at criticality on the complete graph, and the order statistics of the supercritical mesoscopic clusters can be related to the order statistics at the critical point [65, Sect. 1.4.3].
Going beyond the complete graph, is this distribution of ordered cluster sizes universal, at least in sufficiently high dimensions? This would be similar to the conjectured universality of Wigner–Dyson statistics from random matrix theory [67] or the conjectured universality of the distribution of macroscopic loops in loop representations of \(O(n)\) (and other) spin systems [55, 68]. More generally it would be an instance of the universality of low temperature fluctuations in finite volume in models with continuous symmetries.
Finally, we mention that on expander graphs the existence of a phase transition for the arboreal gas is not difficult to show by using a natural split–merge dynamics [54]. It would be interesting if this dynamical approach could also be used to obtain information about the cluster size distribution.
1.4.2 Infinite volume behaviour and relation to the uniform spanning tree
As mentioned previously, the arboreal gas is also known as the uniform forest model [57]. We emphasise that the arboreal gas is not what is typically known as the uniform spanning forest (USF), which is in fact the weak limit as \(\Lambda _{N} \uparrow \mathbb{Z}^{d}\) of a uniform spanning tree (UST) [69]. On a finite graph, the UST is the \(\beta \to \infty \) limit of the arboreal gas. The correct scaling of the external field for this limit is \(h=\beta \kappa \) and we thus write \(\mathbb{P}_{\text{UST},\kappa} = \lim _{\beta \to \infty} \mathbb{P}_{ \beta ,\beta \kappa}\) for the UST on a finite graph (plus ghost vertex if \(\kappa >0\)). For \(\kappa >0\), this measure is also known as the rooted spanning forest, because disregarding the connections to the ghost vertex disconnects the tree of the UST, with vertices previously connected to the ghost becoming roots. The distributions of rooted and unrooted forests are not the same. To help prevent confusion we will refer to the rooted spanning forests as (a special case of) the UST.
It is trivial that \(\mathbb{P}_{\text{UST},0}^{\Lambda _{N}}[0\leftrightarrow x]=1\). Nevertheless, the behaviour of the UST in the weak infinite volume limit depends on the dimension \(d\). This limit can be defined as \(\mathbb{P}_{\mathrm{UST}}^{\mathbb{Z}^{d}} =\lim _{\kappa \downarrow 0} \lim _{N\to \infty} \mathbb{P}_{\text{UST},\kappa}^{\Lambda _{N}}\) and is independent of the finite volume boundary conditions (e.g. free, wired, or periodic as above) imposed on \(\Lambda _{N}\), see [69]. Even though the function \(1_{0\leftrightarrow x}\) is not continuous with respect to the topology of weak convergence, it is still true that
The order of limits here is essential. In this infinite volume limit the UST disconnects into infinitely many infinite trees if \(d>4\), but remains a single connected tree if \(d\leqslant 4\), see [69]. Moreover,
On the left-hand side, the second term vanishes if \(d\leqslant 4\) whereas the first term tends to 0 as \(|x|\to \infty \) if \(d>4\). Furthermore, the geometric structure of the trees under \(\mathbb{P}_{\mathrm{UST}}^{\mathbb{Z}^{d}}\) is well understood. In particular, all trees are one-ended, meaning that removing one edge from a tree results in two trees, of which one is finite [24, 69].
For the arboreal gas, the existence and uniqueness of infinite volume limits is an open question. Nonetheless, subsequential limits exist, and in such an infinite volume limit all trees are finite almost surely when \(\beta \) is small, while Theorem 1.2 implies the existence of an infinite tree for \(\beta \) large. Moreover, by Theorem 1.3,
By analogy with the UST, we expect that only the first term on the left-hand side contributes for \(d\leqslant 4\) and that only the second term contributes asymptotically as \(|x|\to \infty \) for \(d>4\). The tempting conjecture that the UST stochastically dominates the arboreal gas on the torus is consistent with these expectations. The analogue of the left-hand side of (1.31) plays an important role in the proof of uniqueness of the infinite cluster in Bernoulli percolation in [4]; this is related to the vanishing of the second term. As already mentioned, for the arboreal gas we only expect this to be true in \(d\leqslant 4\). Significant progress towards this statement has been obtained in [58], where it is shown that translation-invariant infinite volume limits of the arboreal gas have a unique infinite tree in \(d=3,4\). More precisely, [58] makes use of the existence results of the present paper and establishes uniqueness.
Beyond the questions above, it would be interesting to analyse more detailed geometric aspects of the arboreal gas. For example, can one construct scaling limits as has been done for some spanning tree models [3, 5, 6, 48]?
Finally, we mention that a detailed analysis of the infinite volume behaviour of the arboreal gas on regular trees with wired boundary conditions has been carried out [44, 71]. This infinite volume behaviour is consistent with the finite volume behaviour of the complete graph, e.g., at all supercritical temperatures the sizes of finite clusters have the same distribution as those of critical percolation.
1.4.3 Order of phase transition
Our analysis could be extended to a detailed study of the approach \(h\downarrow 0\). To keep the length of this paper within bounds, we do not carry this out, but here briefly comment on what we expect can be shown by extensions of our analysis. As discussed above, a natural object is the magnetisation
and the corresponding susceptibility (neglecting questions concerning the order of limits)
Thus for the arboreal gas, the susceptibility is not the sum over \(\tau _{\beta ,h}(x)\) as is the case for Bernoulli bond percolation, but the sum over \(\sigma _{\beta ,h}(x)\). In terms of the sigma model, \(\chi \) maybe viewed as the longitudinal susceptibility, often denoted \(\chi _{||}\). In this interpretation, the sum over \(\tau _{\beta ,h}(x)\) is the transversal susceptibility \(\chi _{\perp}\) and satisfies the Ward identity \(\chi _{\perp}(\beta ,h)=\sum _{x} \tau _{\beta ,h}(x) = h^{-1}M( \beta ,h)\) which is crucial in our analysis. For the longitudinal susceptibility, we expect that it would be possible to extend our analysis to show
Defining the free energy \(f(\beta ,h)=\lim _{N\to \infty} |\Lambda _{N}|^{-1}\log Z_{\beta ,h}^{ \Lambda _{N}}\), for \(\beta \geqslant \beta _{0}\) the previous asymptotics suggest that \(h\mapsto f(\beta ,h)\) is \(C^{2}\) in \(d>4\) but only \(C^{1}\) for \(d=3,4\). In fact, extrapolating from our renormalisation group analysis we believe that for \(\beta \geqslant \beta _{0}\) the free energy is \(C^{n}\) but not \(C^{n+1}\) as a function of \(h\geqslant 0\) for \(n=\lfloor{\frac{d-1}{2}} \rfloor \). It is unclear how this is connected to the geometry of the component graph of the UST, which also changes as the dimension is varied [25, 59].
1.4.4 Critical behaviour
The critical behaviour of the \({\mathbb{H} } ^{0|2}\) model and its generalisations (the \({\mathbb{H} } ^{0|2M}\) models) were studied in [46, 62], using \(\varepsilon \)-expansions formally continued from the \(O(n)\) models, with the motivation of being candidates for the CFTs relevant for a dS-CFT correspondence. Rigorous results about the critical behaviour of the arboreal gas on \(\mathbb{Z}^{d}\) for \(d\geqslant 3\) would be very interesting.
1.5 Organisation and notation
This paper is organised as follows. In Sect. 2, we show how Theorem 1.5 is reduced to renormalisation group results with the help of the Ward identity (1.28). The main renormalisation group input is Theorem 2.1 and 2.3. Sections 3–7 then prove these renormalisation group results. Section 3 is concerned with the construction of the bulk renormalisation group flow, and Sect. 4 uses this analysis to compute the susceptibility. In Sect. 5 we extend this construction to include observables. The renormalisation group flow for observables is then used in Sect. 6 to compute pointwise correlation functions. These computations involve a precise analysis of the zero mode. The short Sect. 7 then collects the results and establishes Theorems 2.1 and 2.3. Finally, in Appendix A we collect relations between the arboreal gas and the \(\mathbb{H}^{0|2}\) model as well as basic percolation and high temperature properties of the arboreal gas, and in Appendix B we include some background material about the finite range decomposition that we use.
Throughout we use \(a_{n}\sim b_{n}\) to denote \(\lim _{n\to \infty}a_{n}/b_{n}=1\), \(a_{n}\asymp b_{n}\) to denote the existence of \(c,C>0\) such that \(c a_{n}\leqslant b_{n}\leqslant Ca_{n}\), \(a_{n} \lesssim b_{n}\) if \(a_{n} \leqslant Cb_{n}\), and \(a_{n}=O(b_{n})\) if \(|a_{n}|\lesssim |b_{n}|\). We write \(a\wedge b = \min \{a,b\}\). We consider the dimension \(d \geqslant 3\) to be fixed, and hence allow implicit constants to depend on \(d\). In Sects. 1 and 2 we allow implicit constants to depend on \(L\) as well, as this dependence does not play a role. In subsequent sections \(L\)-dependence is made explicit, though uniformity in \(L\) is only crucial in the contractive estimate of Theorem 3.13. Our main theorems hypothesise \(L=L(d)\) is large, and for geometric convenience we will assume throughout that \(L\) is at least \(2^{d+2}\).
2 Consequences of combining renormalisation and Ward identities
In our renormalisation group analysis, which provides the foundation for the proofs of the theorems stated in Sect. 1, we will not assume any relation between the coupling constants of the quadratic and quartic terms in (1.25) (except that they are small). The equality of the quadratic and quartic couplings is restored with the help of the Ward identity (1.28), i.e.,
This application of the Ward identity is the subject of this section.
In our analysis we distinguish between two orders of limits. We first analyse the ‘infinite volume’ limit \(\lim _{h\downarrow 0} \lim _{N\to \infty}\), and prove Theorem 1.5 (and thus Theorems 1.2–1.3). Using results of this analysis (and with several applications of the Ward identity), we then also analyse the much more delicate ‘finite volume’ limit \(\lim _{N\to \infty}\lim _{h\downarrow 0}\) in order to prove Theorem 1.1.
2.1 Infinite volume correlation functions
For \(m^{2} >0\) arbitrary and coupling constants \(s_{0}\), \(a_{0}\), \(b_{0}\), which eventually will be taken small, we consider the model with fermionic Gaussian reference measure with covariance
on \(\Lambda _{N}\) and interaction
where we recall the squared gradient notation from (1.27). Thus the corresponding expectation is
where \(\int \partial _{\psi }\partial _{\bar{\psi}}\) denotes the Grassmann integral, and \(Z_{m^{2},s_{0},a_{0},b_{0}}\) is defined such that \(\langle 1 \rangle _{m^{2},s_{0},a_{0},b_{0}}=1\). We emphasise the connection to the arboreal gas arises only if \(m^{2}\), \(s_{0}\), \(a_{0}\), \(b_{0}\) are chosen to agree with (1.25), c.f. (2.14)–(2.15) below.
The following result states that for correctly chosen \(a_{0}\) the correlation functions of the fields \(\psi \) and \(\bar{\psi}\psi \) are to leading order multiples of those of the free (Grassmann Gaussian) case \(b_{0}=0\) if first \(N\to \infty \) and then \(m^{2}\downarrow 0\). The result resembles those in [14, 15, 75] for weakly self-avoiding walks in dimension 4. Compared to the latter results, our analysis is substantially simplified since the \(\mathbb{H}^{0|2}\) model can be studied in terms of only fermionic variables with a quartic interaction that is irrelevant in dimensions \(d > 2\). However, in Sect. 2.2, we state an improvement of the following result that captures the full zero mode of the low temperature phase and goes beyond the analysis of [14, 15, 75].
Theorem 2.1
Let \(d \geqslant 3\) and \(L \geqslant L_{0}(d)\). For \(b_{0}\) sufficiently small and \(m^{2} \geqslant 0\), there are \(s_{0} = s_{0}^{c}(b_{0},m^{2})\) and \(a_{0}= a_{0}^{c}(b_{0},m^{2})\) independent of \(N\) so that the following hold: The functions \(s_{0}^{c}\) and \(a_{0}^{c}\) are continuous in both variables, differentiable in \(b_{0}\) with uniformly bounded \(b_{0}\)-derivatives, and satisfy the estimates
uniformly in \(m^{2}\geqslant 0\). There exists \(\kappa >0\) such that if the torus sidelength satisfies \(L^{-N} \leqslant m\),
Moreover, there are functions
having the same continuity properties as \(s_{0}^{c}\) and \(a_{0}^{c}\) such that
Here \(\langle A;B \rangle =\langle AB \rangle -\langle A \rangle \langle B \rangle \).
The proof of this theorem is given in Sects. 3–7. We now show how to derive Theorem 1.5 for the \(\mathbb{H}^{0|2}\) model from it together with the Ward identity (1.28). To this end, assuming \(s_{0}>-1\) we further rescale \(\psi \) by \(1/\sqrt{1+s_{0}}\) (and likewise for \(\bar{\psi}\)) in (1.25), and thus set
Up to a normalisation constant, the fermionic density (1.25) becomes, see also (1.27),
For any \(m^{2} \geqslant 0\) and \(s_{0}>-1\), (2.12) is of the form (2.4) with
To use Theorem 2.1 to study the arboreal gas we need to invert this implicit relation between \((\beta ,h)\) and \((m^{2},s_{0},a_{0},b_{0})\). This is achieved by the following corollary. A key observation is that the Ward identity (1.28) allows us to identify the critical point with \(h=0\). To make this precise, with \(s_{0}^{c}\) and \(a_{0}^{c}\) as in Theorem 2.1, define the functions
By Theorem 2.1, both functions are continuous in \(b_{0}>0\) small enough and \(m^{2} \geqslant 0\).
Corollary 2.2
(i) Assume \(b_{0}>0\) is small enough. Then
In particular, \(h(b_{0},0) = 0\) and \(h(b_{0},m^{2})>0\) if \(m^{2}>0\).
(ii) For \(\beta \) large enough and \(h\geqslant 0\), there are functions \(\tilde{b}_{0}(\beta ,h)>0\) and \(\tilde{m}^{2}(\beta ,h)\geqslant 0\) such that \(h(\tilde{b}_{0},\tilde{m}^{2})=h\) and \(\beta (\tilde{b}_{0},\tilde{m}^{2})=\beta \). Both functions are right-continuous as \(h\downarrow 0\) when \(\beta \) is fixed.
Proof
To prove (i), we use the Ward identity (2.1) with \((\beta ,h)\) given by (2.14)–(2.15). The left- and right-hand sides of (2.1) are, respectively,
By Theorem 2.1, in the limit \(N\to \infty \), we obtain from (2.1) that if \(m^{2}>0\), the identity
holds. Solving for \(h\), we have
Since \(s_{0}^{c}(b_{0},m^{2})=O(b_{0})\), \(\beta (b_{0},m^{2}) \asymp 1/b_{0}\), and \(\gamma (b_{0},m^{2})=O(1)\), all uniformly in \(m^{2}\geqslant 0\), we obtain \(h(b_{0},m^{2}) = m^{2} \beta (b_{0},m^{2})(1+O(b_{0}))\). In particular, \(h(b_{0},0)=0\).
Claim (ii) follows from an implicit function theorem argument that uses that \(s_{0}^{c}\) and \(a_{0}^{c}\) are continuous in \(m^{2}\geqslant 0\) and differentiable in \(b_{0}\) if \(m^{2}>0\) with \(b_{0}\)-derivatives uniformly bounded in \(m^{2}>0\). This argument is the same as the proof of [15, Proposition 4.2] (with our notation \(s_{0}\) instead of \(z_{0}\), \(a_{0}\) instead of \(\nu _{0}\), \(b_{0}\) instead of \(g_{0}\), and with \(1/\beta \) instead of \(g\) and \(h\) instead of \(\nu \)) and is omitted here. □
Assuming Theorem 2.1, the proof of Theorem 1.5 is immediate from the last corollary. The main statements of Theorems 1.2 and 1.3 then follow immediately, except for the identifications \(\theta _{d}(\beta )^{2}=\zeta _{d}(\beta )\), \(({\mathsf{c}}_{2}(\beta )/{\mathsf{c}}_{1}(\beta )^{2})\theta _{d}(\beta )^{2}=1\), and \({\mathsf{c}}(\beta )=2{\mathsf{c}}_{2}(\beta )\) which we will obtain in Sect. 2.2.
Proof of Theorem 1.5
Given \(\beta \geqslant \beta _{0}\) and \(h>0\) we choose \(b_{0}>0\) and \(m^{2}>0\) as in Corollary 2.2 (ii). Since \(z_{x} = 1-\xi _{x}\eta _{x}\) and using (2.11) we then have
Taking \(N\to \infty \) and then \(h\downarrow 0\), the results follow from Corollary 2.2 (i) and Theorem 2.1 with
where the functions \(\lambda \) and \(\gamma \) are evaluated at \(m^{2}=0\) and \(b_{0}\) given as above, \(c_{d}\) is the constant in the asymptotics of the free Green’s function on \(\mathbb{Z}^{d}\), see (1.24), and we have used the simplification of the error terms \(O(|x|^{-(d-2)-1})+O(b_{0} |x|^{-(d-2+\kappa )}) = O(|x|^{-(d-2+ \kappa )})\) and \(O(|x|^{-2(d-2)-1})+O(b_{0} |x|^{-2(d-2)-\kappa )}) = O(|x|^{-2(d-2)- \kappa )})\). □
2.2 Finite volume limit
The next theorem extends Theorem 2.1 by more precise estimates valid in the limit \(m^{2}\downarrow 0\) with \(\Lambda _{N}\) fixed. In these estimates \(t_{N} \in (0,1/m^{2})\) is a continuous function of \(m^{2}>0\) that satisfies
where on the right-hand side \(\Delta ^{\mathbb{Z}^{d}}\) is the Laplacian on \(\mathbb{Z}^{d}\), on the left-hand side \(\Delta \) is the Laplacian on \(\Lambda _{N}\), and \(|\Lambda _{N}| = L^{dN}\) denotes the volume of the torus \(\Lambda _{N}\). We define
so that \(W_{N}(x)\) is essentially the torus Green’s function \((-\Delta +m^{2})^{-1}\) with the zero mode omitted.
In the following theorem, and throughout this section, \(\Lambda _{N}\) is fixed and the parameters \((\beta ,h)\) are related to \((m^{2},s_{0},a_{0},b_{0})\) as in Corollary 2.2. We will write \(\langle \cdot \rangle _{m^{2},b_{0}} = \langle \cdot \rangle _{m^{2},s_{0}^{c}(b_{0},m^{2}),a_{0}^{c}(b_{0},m^{2}),b_{0}}\) for the corresponding expectation and similarly for the partition function \(Z_{m^{2},b_{0}}\).
Theorem 2.3
Under the conditions of Theorem 2.1except that we no longer restrict \(L^{-N}\leqslant m\), in addition to the functions \(a^{c}_{0}\), \(s^{c}_{0}\), \(\lambda \), and \(\gamma \), there are functions \(\tilde{a}_{N,N}^{c} = \tilde{a}_{N,N}^{c}(b_{0},m^{2})\) and \(u_{N}^{c} = u_{N}^{c}(b_{0},m^{2})\), both continuous in \(b_{0}\) small and \(m^{2}\geqslant 0\), as well as
continuous in \(b_{0}\) small and \(m^{2}> 0\), such that, for \(x\in \Lambda _{N}\),
and
The remainder terms satisfy
The proof of this theorem is again given in Sects. 3–7. This proof also gives a bound on \(\tilde{a}_{N,N}^{c}\) of order \(b_{0}L^{-(2+\kappa )N}\) for a small \(\kappa >0\). We did not state this bound above because (by using the Ward identity (2.1)) the existence of \(\tilde{a}_{N,N}^{c}\) with its relation to the correlation functions as stated in the theorem is, in fact, sufficient to determine its precise asymptotic value of order \(b_{0}/|\Lambda _{N}| = b_{0}L^{-dN} \ll b_{0}L^{-(2+\kappa )N}\), see Lemmas 2.4–2.5 below. Using this precise asymptotic information on \(\tilde{a}_{N,N}^{c}\), Theorem 1.1 then follows from Theorem 2.3. The key computation occurs in Lemma 2.6, where the asymptotic value of \(\tilde{a}_{N,N}^{c}\) is used to exhibit important cancelations between the terms on the right-hand side of (2.31).
Lemma 2.4
Under the conditions of Theorem 2.3,
In particular, if \(b_{0}>0\) this implies \(1/\tilde{a}_{N,N}^{c}(b_{0},0) = O(|\Lambda _{N}|/b_{0})\) and \(\tilde{a}_{N,N}^{c}(b_{0},0)>0\).
Proof
From (1.18), we have that
Changing variables,
where \((\beta , h)\) and \((b_{0}, m^{2})\) are related as in (2.14) and (2.15). To evaluate the right-hand side we use (2.29). Note that
where the second equality is due to (2.28) and the third follows from (2.25). As \(m^{2}\downarrow 0\), the right-hand side of the third equality behaves asymptotically as
Since \(s_{0}^{c}(b_{0},0)=O(b_{0})\) by Theorem 2.1 we therefore obtain the first claim:
For the second claim, let us observe that, on the one hand,
where the first equality is by Proposition 1.4 and (2.4), (2.11), and (2.12), and the second equality is (2.32). On the other hand, by (1.1),
Since, by Theorem 2.3, \(u_{N}^{c}\) and \(s_{0}^{c}\) remain bounded as \(m^{2}\downarrow 0\) with \(\Lambda _{N}\) fixed (and thus also \(\beta \) which is given by (2.14)), from \(\det (-\Delta +m^{2}) \downarrow 0\), we conclude that \(1+\tilde{u}_{N,N}^{c}\) diverges as \(m^{2}\downarrow 0\). By (2.28), this implies \(\tilde{a}_{N,N}^{c}(b_{0},0)>0\). The upper bound on \(1/\tilde{a}_{N,N}^{c}(b_{0},0)\) follows by re-arranging (2.35) and using the trivial bound \(|T_{0}| \leqslant |\Lambda _{N}|\). □
Using that \(\tilde{a}_{N,N}^{c}\) is at least of order \(b_{0}/|\Lambda _{N}|\) as established in the previous lemma, the following lemma gives an asymptotic representation of \(\tilde{a}_{N,N}^{c}\) of order \(b_{0}/|\Lambda _{N}|\) in terms of \(\gamma \) from Theorem 2.3.
Lemma 2.5
Under the conditions of Theorem 2.3and if \(b_{0}>0\),
Proof
The Ward identity \(\langle z_{0} \rangle _{\beta ,0}=0\) implies
where we used (2.11) and that \(\beta =\beta (b_{0},m^{2})\) is as in (2.14). To compute \(\langle \bar{\psi}_{0}\psi _{0} \rangle _{m^{2},b_{0}}\), we apply (2.30). Since \(\tilde{u}_{N,N}^{c} = \tilde{a}_{N,N}^{c}t_{N} + O(b_{0}L^{-\kappa N})\) and \(t_{N} = m^{-2}+O(L^{2N})\),
The limits in the second line exist by Theorem 2.3 and Lemma 2.4, which in particular implies \(\tilde{a}_{N,N}^{c}(b_{0},0) > 0\) since \(b_{0}>0\). As \(m^{2}\downarrow 0\), the error term \(E_{00}\) is bounded by \(O(b_{0}L^{-\kappa N}/(|\Lambda _{N}|\tilde{a}_{N,N}^{c})) = (\lambda (b_{0},0) |\Lambda _{N}|^{-1}/\tilde{a}_{N,N}^{c})O(b_{0}L^{-\kappa N})\) since \(\lambda (b_{0},0) =1-O(b_{0})\geqslant 1/2\), finishing the proof. □
Given Theorem 2.3, the following lemma is the main step in the proof of Theorem 1.1. It uses the asymptotic representation of \(\tilde{a}_{N,N}^{c}\) to exhibit cancelations in expressions in Theorem 2.3.
Lemma 2.6
Under the conditions of Theorem 2.3and if \(b_{0}>0\),
where \(\theta _{d}(\beta )\) is defined in (2.24).
Proof
By the last expression for \(\mathbb{P}_{\beta ,0}[0\leftrightarrow x]\) in (1.18) and (2.11), (2.14):
To compute \(\lim _{m^{2}\downarrow 0}\langle \bar{\psi}_{0}\psi _{0}\bar{\psi}_{x} \psi _{x} \rangle _{m^{2},b_{0}}\) we start from (2.31). By Lemma 2.4, as \(m^{2}\downarrow 0\) with \(\Lambda _{N}\) fixed,
This implies the error term in (2.31) is, as \(m^{2}\downarrow 0\) with \(\Lambda _{N}\) fixed,
For the main term we have (recall \(W_{N}(x)=W_{N,m^{2}}(x)\), see (2.27))
where on the right-hand side the functions \(\lambda \), \(\gamma \), and \(\tilde{a}_{N,N}^{c}\) are evaluated at \(m^{2}=0\). By Lemma 2.5,
so that
Substituting these bounds into (2.50) and then (2.47) we obtain
Using the definition (2.24) of \(\theta _{d}(\beta )\), that \(W_{N,0}(x) = (-\Delta ^{\mathbb{Z}^{d}})^{-1}(0,x)+O(L^{-(d-2)N})\) by (2.26), and in particular \(W_{N,0}(x) = O(|x|^{-(d-2)})\), the claim follows. □
The next (and final) lemma is inessential for the main conclusions, but will allow us to identify the constants from the infinite volume and the finite volume analyses.
Lemma 2.7
Under the conditions of Theorem 2.3and if \(b_{0}>0\), then \(\lambda \theta _{d}(\beta ) = 1\).
Proof
Let
where the second equality is due to Lemma 2.4. The density \(\mathbb{E}^{\Lambda _{N}}_{\beta ,0} |T_{0}|/|\Lambda _{N}|\) can also be computed by summing the estimate in Lemma 2.6 and dividing by \(|\Lambda _{N}|\). Subtracting this result from (2.55) gives
On the other hand, (2.43) shows that
The limit \(w= \lim _{N\to \infty} w_{N}\) thus exists and satisfies \(\lambda w = \theta _{d}(\beta )\) and \(w=\theta _{d}(\beta )^{2}\). Since \(\theta _{d}(\beta )=1- O(1/\beta )\neq 0\) this implies \(\lambda \theta _{d}(\beta ) = 1\). □
Proof of Theorem 1.1
The proof follows by rewriting Lemma 2.6. Let \(c_{d}\) be the constant in the Green function asymptotics of (1.24), and recall the constants \(\theta _{d}(\beta )\) and \({\mathsf{c}}_{i}(\beta )\) from (2.24). Theorem 1.1 then follows from Lemma 2.6 by setting
and simplifying the error terms using \(O(b_{0}|x|^{-(d-2)-1})+O(b_{0}^{2} |x|^{-(d-2+\kappa )}) = O(\beta ^{-1}|x|^{-(d-2+ \kappa )})\) and \(O(b_{0}L^{-(d-2)N})+O(b_{0}^{2} L^{-\kappa N}) = O(\beta ^{-1}L^{- \kappa N})\). □
Completion of proof of Theorems 1.2 and 1.3
For Theorem 1.2, \(\zeta _{d}(\beta ) = \theta _{d}(\beta )^{2}\) was established in the previous proof. For Theorem 1.3, the identity \(({\mathsf{c}}_{2}(\beta )/{\mathsf{c}}_{1}(\beta )^{2})\theta _{d}(\beta )^{2}=1\) is equivalent (by (2.24)) to \(\theta _{d}(\beta )\lambda =1\), i.e., Lemma 2.7. Similarly, \({\mathsf{c}}(\beta )=2\lambda \theta _{d}(\beta ) {\mathsf{c}}_{1}(\beta ) = 2{ \mathsf{c}}_{1}(\beta )\). □
Remark 2.8
To compute \(\mathbb{P}_{\beta ,0}[0\leftrightarrow x]\) we started from the expression \(1- \langle \xi _{0}\eta _{0}\xi _{x}\eta _{x} \rangle _{\beta ,0}\) in (1.18). An alternative route would have been to start from \(\langle \xi _{0}\eta _{x} \rangle _{\beta ,0}\). For technical reasons arising in Sect. 5 it is, however, easier to obtain sufficient precision when working with \(\langle \xi _{0}\eta _{0}\xi _{x}\eta _{x} \rangle _{\beta ,0}\).
3 The bulk renormalisation group flow
We will prove Theorems 2.1 and 2.3 by a renormalisation group analysis that is set up following [28, 32] and [14, 15]; see also [18] for a conceptual introduction. Our proof is largely self-contained. The exceptions to self-containment concern general properties about finite range decomposition, norms, and approximation by local polynomials that were developed systematically in [12, 29, 30]. The properties we need are all reviewed in this section. The first six subsections set up the framework of the analysis, and the remaining three define and analyse the renormalisation group flow.
Throughout \(\Lambda =\Lambda _{N}\) is the discrete torus of side length \(L^{N}\). We leave \(L\) implicit; it will eventually be chosen large. We sometimes omit the \(N\) when it does not play a role.
3.1 Finite range decomposition
Let \(\Delta \) denote the lattice Laplacian on \(\Lambda _{N}\), and let \(m^{2}>0\). Our starting point for the analysis is the decomposition
where the \(C_{j}\) (with \(j< N\)) and \(C_{N,N}\) are positive semidefinite \(m^{2}\)-dependent matrices indexed by \(\Lambda _{N}\). These covariances can be chosen with the following properties, see [18, Proposition 3.3.1 and Sect. 3.4] and Appendix B. The notation \(C_{N,N}\) for the last covariance is explained below.
3.1.1 Finite range property
For \(j< N\), the covariances \(C_{j}\) satisfy the finite range property
Moreover, they are invariant under lattice symmetries and independent of \(\Lambda _{N}\) in the sense that \(C_{j}(x,y)\) can be identified as a function of \(x-y\) that is independent of the torus \(\Lambda _{N}\). They are defined and continuous for \(m^{2}\geqslant 0\) including the endpoint \(m^{2}=0\) (and in fact smooth).
3.1.2 Scaling estimates
The covariances satisfy estimates consistent with the decay of the Green function:
where for an arbitrary fixed constant \(s\),
The discrete gradient in (3.3) can act on either the \(x\) or the \(y\) variable, and is defined as follows. Recalling that \(e_{1}, \dots , e_{d}\) denote the standard unit vectors generating \(\mathbb{Z}^{d}\), that \(e_{d+j}=-e_{j}\), and that \(\mathcal {E} _{d} =\{e_{1},\dots ,e_{2d}\}\), for any multiindex \(\alpha \in \mathbb{N}_{0}^{\mathcal {E} _{d}}\), we define the discrete derivative in directions \(\alpha \) with order \(|\alpha |=\sum _{i=1}^{2d} \alpha (e_{i})\) by:
with \(\nabla _{e_{i}}^{k} = \nabla _{e_{i}}\cdots \nabla _{e_{i}}\), where there are \(k\) terms on the right-hand side.
3.1.3 Zero mode
By the above independence of the covariances \(C_{j}\) with \(j< N\) from \(\Lambda _{N}\), all finite volume torus effects are concentrated in the last covariance \(C_{N,N}\). We further separate this covariance into a bounded part and the zero mode:
where \(t_{N}\) is an \(m^{2}\)-dependent constant and \(Q_{N}\) is the projection onto the zero mode, i.e., the matrix with all entries equal to \(1/|\Lambda _{N}|\). The bounded contribution \(C_{N}\) (which does depend on \(\Lambda _{N}\)) satisfies the estimates (3.3) with \(j=N\) and also extends continuously to \(m^{2}=0\). The constant \(t_{N}\) satisfies
In this section, we only consider the effect of \(C_{N}\) (which is parallel to that of the \(C_{j}\) with \(j< N\)) while the nontrivial finite volume effect of \(t_{N}\) will be analysed in Sects. 4–6.
The above properties imply (2.26) and \(W_{N}(x)\) in (2.27) is given by \(W_{N}(x) = C_{1}(x) + \cdots + C_{N}(x)\).
3.2 Grassmann Gaussian integration
For \(X \subset \Lambda =\Lambda _{N}\), we denote by \(\mathcal {N} (X)\) the Grassmann algebra generated by \(\psi _{x}\), \(\bar{\psi}_{x}\), \(x\in X\) with the natural inclusions \(\mathcal {N} (X) \subset \mathcal {N} (X')\) for \(X \subset X'\). Moreover, we denote by \(\mathcal {N} (X \sqcup X)\) the doubled algebra with generators \(\psi _{x}\), \(\bar{\psi}_{x}\), \(\zeta _{x}\), \(\bar{\zeta}_{x}\) and by \(\theta \colon \mathcal {N} (X) \to \mathcal {N} (X \sqcup X)\) the doubling homomorphism acting on the generators of \(\mathcal {N} (X)\) by
For a covariance matrix \(C\) the associated Gaussian expectation \(\mathbb{E}_{C}\) acts on \(\mathcal {N} (X\sqcup X)\) on the \(\zeta \), \(\bar{\zeta}\) variables. Explicitly, when \(C\) is positive definite, \(F\in \mathcal {N} (X\sqcup X)\) maps to \(\mathbb{E}_{C}F \in \mathcal {N} (X)\) given by
Thus \(\mathbb{E}_{C}\theta \colon \mathcal {N} (\Lambda ) \to \mathcal {N} (\Lambda )\) is the fermionic convolution of \(F \in \mathcal {N} (\Lambda )\) with the fermionic Gaussian measure with covariance \(C\). Recall the following well-known facts about \(\mathbb{E}_{C}\theta \); elementary proofs can be found in, e.g., [29]. First, this convolution operator can be written as
where \(\mathcal {L} _{C} = \sum _{x,y\in \Lambda} C_{xy} \partial _{\psi _{y}} \partial _{\bar{\psi}_{x}}\). In particular, it follows that \(\mathbb{E}_{C}\theta \) has the semigroup property
This formula also holds for \(C\) positive semidefinite if we take (3.10) as the definition of \(\mathbb{E}_{C}\theta F\), which we will in the sequel. The identity (3.10) is a fermionic version of the relation between Gaussian convolution and the heat equation, and (3.11), which follows from (3.10), is the analogue of the fact that the sum of two independent Gaussian processes is Gaussian with covariance given by the sum of the covariances. The identity (3.10) allows for the evaluation of moments, e.g., \(\mathbb{E}_{C}\theta \bar{\psi}_{x}\psi _{y}=\bar{\psi}_{x}\psi _{y}+C_{xy}\). An important consequence of the finite range property (3.2) of \(C_{j}\) is that if \(F_{i} \in \mathcal {N} (X_{i})\) with \(\operatorname{dist}_{\infty}(X_{1},X_{2})> \frac{1}{2} L^{j}\) then, by (3.10),
3.3 Symmetries
We briefly discuss symmetries, which are important in extracting the relevant and marginal contributions in each renormalisation group step (see Sect. 3.6 below). We call an element \(F \in \mathcal {N} (\Lambda )\) symplectically invariant or \(U(1)\) invariant if every monomial in its representation has the same number of factors of \(\bar{\psi}\) and \(\psi \). We remark that in [29, 30], to which we will sometimes refer, this property is called (global) gauge invariance. Similarly, \(F \in \mathcal {N} (\Lambda \sqcup \Lambda )\) is \(U(1)\) invariant if the combined number of factors of \(\bar{\psi}\) and \(\bar{\zeta}\) is the same as the combined number of factors of \(\psi \) and \(\zeta \). We denote by \(\mathcal {N} _{\mathrm{sym}}(X)\) the subalgebra of \(\mathcal {N} (X)\) of \(U(1)\) invariant elements and likewise for \(\mathcal {N} _{\mathrm{sym}}(\Lambda \sqcup \Lambda )\). The maps \(\theta \) and \(\mathbb{E}_{C}\) preserve \(U(1)\) symmetry.
A bijection \(E\colon \Lambda \to \Lambda \) is an automorphism of the torus \(\Lambda \) if it maps nearest neighbours to nearest neighbours. Bijections act as homomorphisms on the algebra \(\mathcal {N} (\Lambda )\) by \(E\psi _{x} = \psi _{Ex}\) and \(E\bar{\psi}_{x} = \bar{\psi}_{Ex}\) and similarly for \(\mathcal {N} (\Lambda \sqcup \Lambda )\). If \(C\) is invariant under lattice symmetries, i.e., \(C(Ex,Ey)=C(x,y)\) for all automorphisms \(E\), then the convolution \(\mathbb{E}_{C}\theta \) commutes with automorphisms of \(\Lambda \), i.e., \(E \mathbb{E}_{C}\theta F = \mathbb{E}_{C}\theta E F\). In particular \(E\mathbb{E}_{C_{j}}\theta F = \mathbb{E}_{C_{j}}\theta E F\) for the covariances of the finite range decomposition (3.1). An important consequence of this discussion is that if \(X\subset \Lambda \), \(F\in \mathcal {N} _{\mathrm{sym}}(X)\) and \(F\) is invariant under lattice symmetries that fix \(X\), then \(\mathbb{E}_{C_{j}}\theta F\in \mathcal {N} _{\mathrm{sym}}(X)\) is also invariant under such lattice symmetries.
3.4 Polymer coordinates
We will use (3.11) and the decomposition (3.1) to study the progressive integration
for a given \(Z_{0}\in \mathcal {N} (\Lambda )\). To be concrete here, the reader may keep \(Z_{0}=e^{-V_{0}(\Lambda )}\) with \(V_{0}(\Lambda )\) from (2.3) in mind, but to compute correlation functions we will consider generalisations of this choice of \(Z_{0}\) in Sect. 6. The analysis is performed by defining suitable coordinates (polymer coordinates) and norms (on polymer coordinates) that enable the progressive integration to be treated as a dynamical system: this is the renormalisation group. Towards this end, this section defines local polymer coordinates as in [28, 32]. Section 3.5 then defines relevant norms, and norms on polymer coordinates are introduced in Sect. 3.8 after other preliminary material is introduced.
3.4.1 Blocks and polymers
Recall \(\Lambda =\Lambda _{N}\) denotes a torus of side length \(L^{N}\). Partition \(\Lambda _{N}\) into nested scale-\(j\) blocks \(\mathcal {B} _{j}\) of side lengths \(L^{j}\) where \(j=0,\dots , N\). Thus scale-0 blocks are simply the points in \(\Lambda \), while the only scale-\(N\) block is \(\Lambda \) itself, see Fig. 1. The set of \(j\)-polymers \(\mathcal {P} _{j}=\mathcal {P} _{j}(\Lambda )\) consists of finite unions of blocks in \(\mathcal {B} _{j}\). To define a notion of connectedness, say \(X,Y \in \mathcal {P} _{j}\) do not touch if \(\inf _{x\in X,y\in Y} |x-y|_{\infty }> 1\). A polymer is connected if it is not empty and there is a path of touching blocks between any two blocks of the polymer. The subset of connected \(j\)-polymers is denoted \(\mathcal {C} _{j}\). We will drop \(j\)- prefixes when the scale is clear.
Illustration of \(j\)-blocks when \(L=2\)
For a fixed \(j\)-polymer \(X\), let \(\mathcal {B} _{j}(X)\) denote the set of \(j\)-blocks contained in \(X\) and let \(|\mathcal {B} _{j}(X)|\) be the number of such blocks. Connected polymers \(X\) with \(|\mathcal {B} _{j}(X)| \leqslant 2^{d}\) are called small sets and the collection of all small sets is denoted \(\mathcal {S} _{j}\). Polymers which are not small will be called large. Finally, for \(X \in \mathcal {P} _{j}\) we define its small set neighbourhood \(X^{\square }\in \mathcal {P} _{j}\) as the union of all small sets containing a block in \(\mathcal {B} _{j}(X)\), and its closure \(\bar{X}\) as the smallest \(Y\in \mathcal {P} _{j+1}\) such that \(X\subset Y\).
3.4.2 Coordinates
We will write \(Z_{j}\) in the form
where the \(u_{j}\) are constants (essentially the free energy), the \(V_{j}(X)\) are functions of the fields \(\bar{\psi}_{x}\), \(\psi _{x}\) for \(x\) in a neighbourhood of \(X\), parametrised by finitely many coupling constants which require special attention (and are independent of \(X\)), and everything else is organised into the functions \(K_{j}(X)\), which will be called polymer activities. Unlike the \(V_{j}\), the polymer activities track quantities whose precise value is not important. Explicit (somewhat complicated) formulas for the evolution of \(K_{j}\) will be given below. An essential point will be that they can be tracked in terms of estimates. The tuple \((V_{j},K_{j})\) together with the representation (3.14) will be referred to as polymer coordinates. In the remainder of this subsection, we will discuss some structural properties of these coordinates.
Coupling constants. We will always identify \(V_{j}\) with the coupling constants which parametrise it. Explicitly, for coupling constants \(V_{j}=(z_{j},y_{j},a_{j},b_{j}) \in \mathbb{C}^{4}\) and a set \(X \subset \Lambda _{N}\), let
For the scale \(j=0\), if we set \(Z_{0} = e^{-V_{0}(\Lambda _{N})}\), then the polymer coordinates take the simple form
with
To study the recursion \(Z_{j+1}=\mathbb{E}_{C_{j+1}}\theta Z_{j}\) at a general scale \(j=1,\dots ,N\), we will make a choice of coupling constants \(V_{j}\) and of polymer activities \(K_{j}=(K_{j}(X))_{X\in \mathcal {P} _{j}(X)}\) such that
Polymer activities. The \(K_{j}\) will be defined in such a way that they satisfy the locality and symmetry property \(K_{j}(X) \in \mathcal {N} _{\mathrm{sym}}(X^{\square})\) and the following important component factorisation property: for \(X,Y\in \mathcal {P} _{j}\) that do not touch,
Note that since they are \(U(1)\) symmetric, the \(K_{j}(X)\) are even elements of \(\mathcal {N} \), so they commute and the product on the right-hand side is unambiguous. Using the previous identity,
where \(\operatorname{Comp}(X)\) denotes the set of connected components of the polymer \(X\). In particular, each \(K_{j}= (K_{j}(X))_{X\in \mathcal {P} _{j}(\Lambda _{N})}\) satisfying (3.19) can be identified with its restriction \(K_{j}= (K_{j}(X))_{X\in \mathcal {C} _{j}(\Lambda _{N})}\). We say that \(K_{j}\) is automorphism invariant if \(EK_{j}(X) = K_{j}(E(X))\) for all \(X\in \mathcal {P} _{j}(\Lambda _{N})\) and all torus automorphisms \(E \in {\mathrm{Aut}}(\Lambda _{N})\) that map blocks in \(\mathcal {B} _{j}\) to blocks in \(\mathcal {B} _{j}\).
Definition 3.1
Let \(\mathcal {K} ^{\varnothing }_{j}(\Lambda _{N})\) be the linear space of automorphism invariant \(K_{j} = (K_{j}(X))_{X\in \mathcal {C} _{j}(\Lambda _{N})}\) with \(K_{j}(X) \in \mathcal {N} _{\mathrm{sym}}(X^{\square})\) for every \(X \in \mathcal {C} _{j}\).
Polymer coordinates at scale \(j\) are thus a choice of (the coupling constants) \(V_{j}\) together with a choice of (polymer activities) \(K_{j}\) from the space \(\mathcal {K} ^{\varnothing }_{j}\). The renormalisation group map is a particular choice of a map \((V_{j},K_{j})\mapsto (V_{j+1},K_{j+1})\).
For a given \(Z_{j}\), the above conditions do not determine \(K_{j}\) uniquely given \(V_{j}\) (see the proof of Proposition 3.11, where the non-uniqueness is apparent). We will state our specific choice of such a map in Sect. 3.7 below. The goal is to choose \(V_{j}\) such that the size of the \(K_{j}\) decrease rapidly as \(j\) increases when the sizes of \(V_{j}\) and \(K_{j}\) are measured in appropriate norms. Thus \(K_{j}\) will capture the irrelevant (or contracting) directions of the renormalisation group dynamics, while the relevant (or expanding) and marginal directions will be captured by the \(V_{j}\) coordinates. The next section defines the norms we will use.
3.5 Norms
We now define the \(T_{j}(\ell )\) norms we will use on the Grassmann algebras \(\mathcal {N} (\Lambda )\). General properties of these norms were systematically developed in [29], to which we will refer for some proofs. To help the reader, in places where we specialise the definitions of [29] we indicate the more general notation that is used in [29].
We start with some notation. For any set \(S\), we write \(S^{*}\) for the set of finite sequences in \(S\). We write \(\Lambda _{f} = \Lambda \times \{\pm 1\}\) and for \((x,\sigma ) \in \Lambda _{f}\) we write \(\psi _{x,\sigma} = \psi _{x}\) if \(\sigma =+1\) and \(\psi _{x,\sigma}=\bar{\psi}_{x}\) if \(\sigma =-1\). Then every element \(F \in \mathcal {N} (\Lambda )\) can be written in the form
where \(\psi ^{z} = \psi _{z_{1}}\cdots \psi _{z_{n}}\) if \(z=(z_{1},\dots , z_{n})\). We are using the notation that \(z!=n!\) if the sequence \(z\) has length \(n\). The representation in (3.21) is in general not unique. To obtain a unique representation we require that the \(F_{z}\) are antisymmetric with respect to permutations of the components of \(z\) (this is possible due to the antisymmetry of the Grassmann variables). Antisymmetry implies that \(F_{z}=0\) if \(z\) has length exceeding \(2|\Lambda |\) or if \(z\) has any repeated entries.
Definition 3.2
Let \(p_{\Phi}=2d\). The space of test functions \(\Phi _{j}(\ell )\) is defined as the set of functions \(g\colon \Lambda _{f}^{*} \to \mathbb{R}\), \(z\mapsto g_{z}\) together with norm
In this definition, \(\nabla _{z_{i}}^{\alpha _{i}}\) denotes the discrete derivative \(\nabla ^{\alpha _{i}}\) with multiindex \(\alpha _{i}\) acting on the spatial part of the \(i\)th component of the finite sequence \(z\).
The \(\Phi _{j}(\ell )\) norm measures spatial smoothness of test functions, which act as substitutes for fields. Restricted to sequences of fixed length, it is a lattice \(C^{p_{\Phi}}\) norm at spatial scale \(L^{j}\) and field scale \(\ell \). We will mainly use the following choice of \(\ell \) when using the \(\Phi _{j}(\ell )\) norm:
for a large constant \(\ell _{0}\), and \(\ell _{j}\) will always be as in (3.23). This choice captures the size of the covariances in the decomposition (3.1). Indeed, regarding the covariances \(C_{j}\) as functions of sequences of length 2 (i.e., as the coefficient in (3.21) of \(F= \sum _{x,y} \bar{\psi}_{x}\psi _{y}C_{j}(x,y)\)), the bounds (3.3) imply
when \(\ell _{0}\) is chosen as a large (\(L\)-dependent, due to the index \(j+1\) on the left-hand side of (3.3)) constant relative to the constants in (3.3) with \(|\alpha |\leqslant 2p_{\Phi}\). From now on, we will always assume that \(\ell _{0}\) is fixed in this way.
Definition 3.3
We define \(T_{j}(\ell )\) to be the algebra \(\mathcal {N} (\Lambda )\) together with the dual norm
when \(F\in \mathcal {N} (\Lambda )\) is expressed as in (3.21).
An analogous definition applies to \(\mathcal {N} (\Lambda \sqcup \Lambda )\), and we then write \(T_{j}(\ell \sqcup \ell ) =T_{j}(\ell )\) for this norm (with the first notation to emphasise the doubled algebra), where we recall that \(\mathcal {N} (\Lambda \sqcup \Lambda )\) is defined above (3.8).
The \(T_{j}(\ell )\) norm measures smoothness of field functionals \(F\in \mathcal {N} (\Lambda )\) with respect to fields whose size is measured by \(\Phi _{j}(\ell )\). They therefore implement the power counting on which renormalisation relies. Important, but relatively straightforwardly verified, properties of these norms are systematically developed in [29]; we summarise the ones we need now.
Product property. First, the \(T_{j}(\ell )\) norm defines a Banach algebra, i.e., the following product property holds (see [29, Proposition 3.7]): for \(F_{1},F_{2} \in \mathcal {N} (\Lambda )\),
Using the product property, we may gain some intuition regarding these norms by considering the following simple examples:
The following more subtle example relies on \(\psi _{x}^{2}=\bar{\psi}_{x}^{2}=0\) and plays an important role for our model:
In general each factor of the fields contributes a factor \(\ell \) and each derivative a factor \(L^{-j}\).
Monotonicity. Second, as follows immediately from the definition, the following monotonicity properties hold: for \(\ell \leqslant \ell '\) and \(F \in \mathcal {N} (\Lambda )\),
Doubling map. Third, the doubling map satisfies (see [29, Proposition 3.12]): for \(F \in \mathcal {N} (\Lambda )\),
where the norm on the left-hand side is the \(T_{j}(\ell ) =T_{j}(\ell \sqcup \ell )\) norm on \(\mathcal {N} (\Lambda \sqcup \Lambda )\).
Gram inequality. Finally, the following contraction bound for the fermionic Gaussian expectation is an application of the Gram inequality whose importance is well-known in fermionic renormalisation. It is proved in [29, Proposition 3.19].
Proposition 3.4
Assume \(C\) is a covariance matrix with \(\|C\|_{\Phi _{j}(\ell )} \leqslant 1\). For \(F \in \mathcal {N} (\Lambda \sqcup \Lambda )\), then
In particular, for \(F\in \mathcal {N} (\Lambda )\), by (3.31) the fermionic Gaussian convolution satisfies
For our choices of \(\ell _{j}\) and of the finite range covariance matrices \(C_{j}\), the inequalities (3.30) and (3.33) in particular imply
We remark that the existence of this contraction estimate for the expectation combined with (3.40) below is what makes renormalisation of fermionic fields much simpler than that of bosonic ones.
3.6 Localisation
To define the renormalisation group map we need one more important ingredient: the localisation operators \(\operatorname{Loc}_{X}\) and \(\operatorname{Loc}_{X,Y}\) that will be used to extract the relevant and marginal terms from the \(K_{j}\) coordinate to incorporate them in the renormalisation from \(V_{j}\) into \(V_{j+1}\). These operators are generalised Taylor approximations which take as inputs \(F\in \mathcal {N} (X)\) and produce best approximations of \(F\) in a finite dimensional space of local field polynomials.
3.6.1 Local field polynomials
By formal local field polynomials we refer to formal polynomials in the symbols \(\psi ,\bar{\psi},\nabla \psi ,\nabla \bar{\psi},\Delta \psi ,\Delta \bar{\psi},\nabla ^{2}\psi ,\dots \) (without spatial index). The dimension of a formal local field monomial is given by \((d-2)/2\) times the number of factors of \(\psi \) or \(\bar{\psi}\) plus the number of discrete derivatives \(\nabla \) in its representation, where \(\Delta \) is treated as two discrete derivatives. The classification of local monomials according to dimension is known as power counting in the renormalisation group literature. Relevant monomials are those with dimension strictly greater than \(d\), marginal ones those with dimension equal to \(d\), and irrelevant those with dimension strictly less than \(d\). Concretely, we consider the following space of formal local field polynomials, consisting of the relevant and marginal monomials consistent with symmetry constraints.
Definition 3.5
Let \(\mathcal {V} ^{\varnothing }\cong \mathbb{C}^{4}\) be the linear space of formal local field polynomials of the form
We will identify elements \(V\in \mathcal {V} ^{\varnothing }\) with their coupling constants \((z,y,a,b) \in \mathbb{C}^{4}\). Sometimes we include a constant term \(u\) and write \(u+V \in \mathbb{C}\oplus \mathcal {V} ^{\varnothing }\) with \(u+V\cong (u,z,y,a,b)\in \mathbb{C}^{5}\).
Given a set \(X\subset \Lambda \), a formal local field polynomial \(P\) can be specialised to an element of \(\mathcal {N} (\Lambda )\) by replacing formal monomials by evaluations. For example, if \(P=\bar{\psi}\psi \), \(P(X) = \sum _{x\in X}\bar{\psi}_{x}\psi _{x}\). We call polynomials arising in this way local polynomials. The most important case is \(V \mapsto V(X)\), with
where \(\Delta =-\frac{1}{2} \sum _{e\in \mathcal {E} _{d}} \nabla _{-e}\nabla _{e}\) and \((\nabla \psi )_{x}(\nabla \bar{\psi})_{x} =\frac{1}{2} \sum _{e\in \mathcal {E} _{d}}\nabla _{e}\psi _{x}\nabla _{e}\bar{\psi}_{x}\) are the lattice Laplacian and the square of the lattice gradient; recall that \(\mathcal {E} _{d} = \{e_{1}, \dots , e_{2d}\}\). For a constant \(u\in \mathbb{C}\) we write \(u(X) = u|X|\), where \(|X|\) is the number of points in \(X \subset \Lambda \). Thus \((u+V)(X)= u(X)+V(X)=u|X|+V(X)\).
Definition 3.6
For \(X\subset \Lambda \), define \(\mathcal {V} ^{\varnothing }(X) = \{V(X): V\in \mathcal {V} ^{\varnothing }\} \subset \mathcal {N} (\Lambda )\) and analogously \((\mathbb{C}\oplus \mathcal {V} ^{\varnothing })(X) = \{ u|X|+V(X): u\in \mathbb{C}, \, V \in \mathcal {V} ^{\varnothing }\} \subset \mathcal {N} (\Lambda )\).
The space \(\mathcal {V} ^{\varnothing }\) contains all formal local field polynomials whose constituent monomials have dimension at most \(d\) that are (i) \(U(1)\) invariant, (ii) respect lattice symmetries (if \(EX=X\) for an automorphism \(E\), then \(EV(X)=V(X)\)), (iii) \(V(X)\neq 0\), and (iv) have no constant terms. Note that \(\mathbb{E}_{C}\theta \) preserves \(\mathcal {V} ^{\varnothing }(X)\) by the discussion in Sect. 3.3. We emphasise that there is no \((\bar{\psi}\psi )^{2}\) term, which would be consistent with having dimension as most \(d\) (if \(d=3,4\)) and symmetries, because it vanishes upon specialisation by anticommutativity of the fermionic variables.
Two further remarks are in order. First, the monomial \(\psi \bar{\psi}(\nabla \psi )(\nabla \bar{\psi})\) has dimension \(2d-2>d\) for \(d\geqslant 3\); we include it in \(\mathcal {V} ^{\varnothing }\) since it occurs in the initial potential. Second, the monomials multiplying \(z\) and \(y\) are equivalent upon specialisation when \(X=\Lambda \) by summation by parts, and differ only by boundary terms for general \(X \subset \Lambda \). This would allow us to keep only one of them, but it will be simpler to keep both.
3.6.2 Localisation
The localisation operators \(\operatorname{Loc}_{X}\) and \(\operatorname{Loc}_{X,Y}\) associate local field monomials to elements of \(\mathcal {N} (X)\). In renormalisation group terminology, the image of \(\operatorname{Loc}\) projects onto the space of all relevant and marginal local polynomials. The precise definitions of the localisation operators do not play a direct role in this paper. Rather, only their abstract properties, summarised in the following Proposition 3.8, will be required. Nonetheless, to give some intuition for the action of \(\operatorname{Loc}_{X}\) and \(\operatorname{Loc}_{X,Y}\), we include the following typical examples (see also [30, Sect. 1.5]). The examples indicate (as stated at the beginning of this section) that the localisation operators are generalised Taylor approximations.
Example 3.7
(i) Let \(F\) be a monomial in \(\{\bar{\psi}_{x}, \psi _{x}\}\) of degree greater than four. Then \(\operatorname{Loc}_{X} F = 0\).
(ii) Consider \(F= \sum _{x\in X}\sum _{y\in \Lambda} q(x-y)\bar{\psi}_{y}\psi _{y}\) where the kernel \(q\colon \mathbb{Z}^{d} \to \mathbb{R}\) has finite support and is invariant under lattice rotations. Then provided \(\Lambda \) is large enough,
where \(q^{(1)}= \sum _{y \in \mathbb{Z}^{d}} q(y)\) and \(q^{(**)} = \sum _{y \in \mathbb{Z}^{d}} y_{1}^{2} q(y)\) (and \(y_{1}\) denotes the first component of \(y\in \mathbb{Z}^{d}\)), and with the same \(P_{x}\) as in (3.37),
Thus \(\sum _{i=1}^{n} \operatorname{Loc}_{X,X_{i}} F =\operatorname{Loc}_{X} F\) if \(X\) is the disjoint union of \(X_{1},\dots , X_{n}\).
For the definition of \(\operatorname{Loc}_{X}\) and \(\operatorname{Loc}_{X,Y}\), we use the general framework developed in [30]. In short, the definitions of \(\operatorname{Loc}_{X}\) and \(\operatorname{Loc}_{X,Y}\) are those of [30, Definition 1.6 and 1.15]. These definitions require a choice of field dimensions, which we choose as \([\psi ]=[\bar{\psi}] = (d-2)/2\), a choice of maximal field dimension \(d_{+}\), which we choose as \(d_{+}=d\), and a choice of a space \(\hat{P}\) of test polynomials, which we define exactly as in [30, (1.19)] with the substitution \(\nabla _{e}\nabla _{e}\to -\nabla _{e}\nabla _{-e}\) explained in [30, Example 1.3]. The following properties are then almost immediate from [30].
Proposition 3.8
For \(L=L(d)\) sufficiently large there is a universal \(\bar{C}>0\) such that: for \(j< N\) and any small sets \(Y \subset X \in \mathcal {S} _{j}\), the linear maps \(\operatorname{Loc}_{X,Y}\colon \mathcal {N} (X^{\square}) \to \mathcal {N} (Y^{\square})\) have the following properties:
(i) They are bounded:
(ii) The maps \(\operatorname{Loc}_{X} =\operatorname{Loc}_{X,X}\colon \mathcal {N} (X^{\square}) \to \mathcal {N} (X^{ \square})\) satisfy the contraction bound
(iii) If \(X\) is the disjoint union of \(X_{1}, \dots , X_{n}\) then \(\operatorname{Loc}_{X} = \sum _{i=1}^{n} \operatorname{Loc}_{X,X_{i}}\).
(iv) The maps are Euclidean invariant: if \(E\in {\mathrm{Aut}}(\Lambda _{N})\) then \(E \operatorname{Loc}_{X,Y} F = \operatorname{Loc}_{EX,EY} EF\).
(v) For a block \(B\), small polymers \(X_{1},\dots , X_{n}\), and any \(F_{i} \in \mathcal {N} _{\mathrm{sym}}(X_{i}^{\square})\) such that \(\sum _{i=1}^{n}\operatorname{Loc}_{X_{i},B} F_{i}\) is invariant under automorphisms of \(\Lambda _{N}\) that fix \(B\),
We remark that the image of \(\operatorname{Loc}_{X,Y}\) is in general a larger space of local field monomials than \(\mathcal {V} ^{\varnothing }(Y)\), often denoted \(\mathcal {V} \) in [30] — for example first gradients of the field can arise which only need cancel upon the symmetrisation in (3.41). Since we will not use this larger space directly we have not assigned a symbol for it.
Proof of Proposition 3.8
The bound (i) is [30, Proposition 1.16], the contraction bound (ii) is [30, Proposition 1.12], the decomposition property (iii) holds by the definition of \(\operatorname{Loc}_{X,Y}\) in [30, Definition 1.15], and the Euclidean invariance (iv) is [30, Proposition 1.9]. Note that the parameter \(A'\) in [30, Proposition 1.12] does not appear here as it applies to the boson field \(\phi \); our fermionic context corresponds to \(\phi =0\). For the application of [30, Proposition 1.12] we have used that \(p_{\Phi}\) was fixed to be \(2d\) in Definition 3.2, and that we have only considered the action of \(\operatorname{Loc}\) on small sets.
Finally, property (v) follows from [30, Proposition 1.10] and the fact that the space \(\mathcal {V} ^{\varnothing }\) defined in Definition 3.5 contains all local polynomials of dimension at most \(d\) invariant under lattice automorphisms that fix a point. □
3.7 Definition of the renormalisation group map
The renormalisation group map \(\Phi _{j+1} = \Phi _{j+1,N,m^{2}}\) is a map
acting on
with \(u_{j+1}\in \mathbb{C}\), the space of coupling constants \(\mathcal {V} ^{\varnothing }\) as in Definition 3.5, and the space of polymer activities \(\mathcal {K} ^{\varnothing }_{j}(\Lambda _{N})\) as in Definition 3.1. The map will have mild dependence on \(m^{2}\) and \(\Lambda _{N}\) as a consequence of this dependence of the covariance matrices. As indicated above the \(u\)-coordinate does not influence the dynamics of the remaining coordinates. Thus we can always explicitly assume that the incoming \(u\)-component of \(\Phi _{j+1}\) is 0 and separate it from \(V_{j+1}\) in the output. This means that we will often regard \(\Phi _{j+1}\) as a map \((V_{j},K_{j}) \mapsto (V_{j+1},K_{j+1})\) where \(u_{j}=u_{j+1}=0\).
The explicit definition of the map \(\Phi _{j+1}\) is given in (3.48) and (3.49) below. The essential consequences of the definition are Proposition 3.11, which enables the iterative application of the renormalisation group maps, and the estimates of Theorem 3.13.
At first sight, the definition of \(\Phi _{j+1}\) may appear somewhat complicated, but it follows from simple principles that are outlined in the proof of Proposition 3.11 below. Compared to other implementations of the fermionic renormalisation group, the finite range property of the covariances in our implementation means we do not require infinite expansions, nor do we require norms which control the spatial complexity of polymers beyond simple volume estimates. As such, establishing useful norm estimates becomes an essentially combinatorial problem. This feature is especially useful in models with bosonic fields, see [28, 36] and [18, Appendix A] for introductory discussions, but it also provides appealing features in the present fermionic context. For example, the flows on tori with two distinct side lengths \(L^{N_{1}}< L^{N_{2}}\) coincide up to the final length scale \(L^{N_{1}-1}\) for polymers which do not wrap around either torus, making the definition of the infinite volume flow and its relation to the finite volume one particularly transparent (see Proposition 3.12 below).
For the definition of the renormalisation group map \(\Phi _{j+1}\), we identify \(V_{j} \in \mathcal {V} ^{\varnothing }\) with the tuple \((V_{j}(B))_{B \in \mathcal {B} _{j}(\Lambda _{N})}\), i.e., the field monomials corresponding to the coupling constants \(V_{j}\) evaluated over a block \(B\), and the tuple \((K_{j}(X))_{X\in \mathcal {C} _{j}(\Lambda _{N})}\) with its extension \((K_{j}(X))_{X\in \mathcal {P} _{j}(\Lambda _{N})}\) determined by the component factorisation property (3.19). We also introduce, assuming \(j+1< N\),
and \(J(B,X)=0\) otherwise. If \(j+1=N\) we simply set \(Q=J=0\).
As a consequence of the properties of \(\operatorname{Loc}\) from Proposition 3.8 (c.f. in particular property (v)), \(Q(B)\) arises from an element of \(\mathcal {V} ^{\varnothing }\) and represents the marginal and relevant contributions from \(K_{j}\) associated with the block \(B\). These contributions (which are marginal or relevant in the sense of power counting) only come from \(K(X)\) for small sets \(X\), as large sets \(X\) will yield contracting contributions for entropic reasons (as opposed to power counting reasons) considered later. The \(J(B,X)\) are a technical device for removing the \(Q\) contribution from \(K_{j}\). An important property is that
The application of this property occurs in (3.111). We indicate the motivation for the form of \(J\) below. For a fuller discussion we refer to [28, Lectures 4–5] and [19, Appendix A, 12.3.2].
We will specify the \(V\)- and \(K\)-components of the renormalisation group map separately.
\(V\)-component. The first definition defines the \(V\)-component of the renormalisation group map. This map is given by first-order perturbation theory, i.e., \(\mathbb{E}_{C_{j+1}}[\theta V_{j}(B)]\), plus the higher-order contribution \(\mathbb{E}_{C_{j+1}}[\theta Q(B)]\) representing the marginal and relevant contributions from \(K_{j}\) as discussed above. Here recall the definition of the doubling map \(\theta \) from (3.8), i.e., \(\theta F\) is obtained from \(F\) by replacing \(\psi \) by \(\psi +\zeta \) and \(\bar{\psi}\) by \(\bar{\psi}+\bar{\zeta}\), and that the expectation only acts on \((\zeta ,\bar{\zeta})\).
Definition 3.9
The map \((V_{j},K_{j}) \mapsto (u_{j+1},V_{j+1})\) is defined by
We emphasise that \(V_{j+1}\) is evaluated on \(B\in \mathcal {B} _{j}\) here; \(V_{j+1}\) can then be extended to \(\mathcal {B} _{j+1}\) by additivity. When \(K_{j}\) is automorphism invariant, which is the case if \(K_{j}\in \mathcal {K} ^{\varnothing }_{j}(\Lambda _{N})\), the right-hand side of (3.48) is in \((\mathbb{C}\oplus \mathcal {V} ^{\varnothing })(B)\) and can thus be identified with an element of \(\mathbb{C}\oplus \mathcal {V} ^{\varnothing }\cong \mathbb{C}^{5}\). This can be checked by using Proposition 3.8 (iv) and (v) and the properties of progressive integration discussed in Sect. 3.2. Recall that we sometimes write the left-hand side as \((u+V)_{j+1}(B)\). Since \(V_{j+1}(B)\) has no constant term by definition, the constant \(u_{j+1}\) is unambiguously defined.
K-component. The following formula for the \(K\)-component of the renormalisation group map is more involved. It is engineered to achieve the desired factorisation and contraction properties of the renormalisation group map. The explicit formula will enable a relatively straightforward verification of the estimates which follow from it; the formula itself is the result of relatively simple manipulations explained in the proof Proposition 3.11 below.
Definition 3.10
For \(U \in \mathcal {P} _{j+1}\), the map \((V_{j},K_{j}) \mapsto K_{j+1}(U)\) is defined by
where
Following [28, Sect. 5.1.2], we define the set \(\mathcal {G} (U)\) (and the corresponding notation \(\mathcal {X} \) and \(X_{\mathcal {X} }\)) as follows: \(\check{X}\in \mathcal {P} _{j}\) and \(\mathcal {X} \) is a set of pairs \((B,X)\) with \(X \in \mathcal {S} _{j}\) and \(B \in \mathcal {B} _{j}(X)\) with the following properties: each \(X\) appears in at most one pair \((B,X)\in \mathcal {X} \), the different \(X\) do not touch, \(X_{\mathcal {X} }= \cup _{(B,X)\in \mathcal {X} } X\) does not touch \(\check{X}\), and the closure of the union of \(\check{X}\) with \(\cup _{(B,X)\in \mathcal {X} } B^{\square}\) is \(U\).
The following proposition is essentially [28, Proposition 5.1]. The only differences are that we have factored out the factor \(e^{-u_{j+1}|\Lambda |}\) and that the doubling map \(\theta \) is explicit (it is implicit in [28]). Explicitly, note that \(K_{j+1}(U)\) and \(V_{j+1}(X)\) are elements of the Grassmann algebra \(\mathcal {N} (\Lambda )\), i.e., they depend on the fields \((\psi ,\bar{\psi})\), but since the doubling map \(\theta \) replaces \((\psi ,\bar{\psi})\) by \((\psi +\zeta ,\bar{\psi}+\bar{\zeta})\), the functions \(\check{K}_{j}(X)\) and \((\delta I)^{X}\) above depend on all of \((\psi ,\bar{\psi},\zeta ,\bar{\zeta})\).
For convenience and because it also demystifies the somewhat complicated formula for the renormalisation group map, we have included a proof along with some additional explanations. The proof of this proposition does not rely on the specific choice of \(Q\) and \(J\) in (3.44)–(3.46) or on the property (3.47). This choice only becomes important in the proof of Theorem 3.13.
Proposition 3.11
Given \((V_{j},K_{j})\) define \(Z_{j}\) by (3.18) with \(u_{j}=0\). Suppose \(K_{j}\) has the factorisation property (3.19) with respect to \(\mathcal {P} _{j}\). Then with the above choice of \((u_{j+1},V_{j+1},K_{j+1})\) and \(Z_{j+1}\) given by (3.18) with \(j+1\) in place of \(j\), we have \(Z_{j+1}=\mathbb{E}_{C_{j+1}}\theta Z_{j}\), and \(K_{j+1}\) has the factorisation property (3.19) with respect to \(\mathcal {P} _{j+1}\). Moreover, if \(K_{j}\) is automorphism invariant then so is \(K_{j+1}\).
Proof
The proof essentially consists of algebraically manipulating the expression
where \(I(B) = e^{-V_{j}(B)}\) and \(K(X)= K_{j}(X)\). These manipulations only rely on factorisation properties of \(I\) and \(K\) (and not on their precise definitions). Hence we will explicitly state the required factorisation properties, and later specialise to the context of Proposition 3.11. We will use that \(I^{Y} = \prod _{B \in \mathcal {B} _{j}(Y)} I(B)\) factors over blocks and \(K(X)\) factors over connected components of \(X\).
Change of coupling constants. Given any \(\tilde{I}(B) \in \mathcal {N} (B)\) for \(B \in \mathcal {B} _{j}\), let \(\delta I(B) = \theta I(B)-\tilde{I}(B)\) and \(\tilde{I}^{Y}=\prod _{B\in \mathcal {B} _{j}(Y)}\tilde{I}(B)\), i.e., \(\theta I(B)\) depends on \((\psi +\zeta ,\bar{\psi}+\bar{\zeta})\) by definition of \(\theta \), \(\tilde{I}(B)\) on \((\psi ,\bar{\psi})\), and \(\delta I(B)\) depends on \((\psi ,\bar{\psi},\zeta ,\bar{\zeta})\). The binomial expansion identity
and (3.52) lead to, after changing the index of summation,
We will later make the particular choice of \(\tilde{I}\) that corresponds to (3.51). Thus \(\tilde{I}\) corresponds to the next-scale coupling constants \(V_{j+1}\). This will be important for obtaining the desired contractive properties of the renormalisation group map in Theorem 3.13. Exhibiting that the \(K\) coordinate is contractive will be aided by the following re-arrangements.
Cancellation of small sets. Keeping in mind that \(\tilde{K}\) factors over components (since \(K\) factors over components), we can then define \(\check{K}(Y)\) to be \(\tilde{K}(Y) - \sum _{B\in \mathcal {B} _{j}(Y)}\theta J(B,Y)\) for any connected polymer \(Y\) (and zero otherwise), where \(J(B,Y)\in \mathcal {N} (B)\) are given. This yields the following formula for \(\tilde{K}\):
Again we will later specialise to \(J\) as defined in (3.45) and (3.46), in particular \(J(B,Y) = 0\) unless \(Y\) is a small set. The motivation for this step is that, for \(Y\) that are small sets but not blocks, the effect is that \(\check{K}\) has the relevant and marginal contributions corresponding to \(\sum _{B} J(B,Y)\) removed. This step does not remove the relevant and marginal contributions of blocks due to the choice (3.45). However, relevant and marginal contributions of blocks will be removed by appropriate choice of \(\tilde{I}\). Both cancellations occur in the estimates in Sect. 3.8.3.
We next substitute (3.55) into (3.54) and re-arrange the resulting sum. Expanding the product, (3.55) can be written as
Given the polymer \(X \setminus \check{X}\), there is a \((X\setminus \check{X})\)-dependent set of sets \(\mathcal {X} \subset \{(B,Y): B\in \mathcal {B} _{j}(Y), Y\in \mathcal {C} _{j}\}\) such that
where the sum on the right-hand is over the aforementioned sets of \(\mathcal {X} \). Explicitly, the sets comprising \(\mathcal {X} \) are sets of pairs \((B,Y)\) where (i) \(B\) is a block in \(Y\), (ii) each component \(Y\) occurs in exactly one pair, and (iii) \(Y\) is a component of \(X\setminus \check{X}\). In particular, \(X = \check{X}\cup X_{\mathcal {X} }\). Thus
Substituting this into (3.54), and using that \(\tilde{I}^{X} \in \mathcal {N} (\Lambda )\) is a constant with respect to \(\mathbb{E}_{C_{j+1}}\), i.e., the expectation acts on \((\zeta ,\bar{\zeta})\) while \(\tilde{I}^{X}\) depends on \((\psi ,\bar{\psi})\),
where \(X_{\mathcal {X} }\) is by definition \(\bigcup _{(B,Y)\in \mathcal {X} }Y\), and hence \(X=\check{X}\cup X_{\mathcal {X} }\) by the definition of the set \(\mathcal {X} \).
Reblocking. Next we organise the last right-hand side of (3.59) as a sum over next-scale polymers \(U\in \mathcal {P} _{j+1}\). We start by inserting the partition of unity
into the sum and changing the order of the sums. This gives
with
where we make the definition that \(\mathcal {G} (U)\) consists of \((\mathcal {X} ,\check{X})\) such that \(\check{X}\in \mathcal {P} _{j}\), \(\mathcal {X} \) satisfies (i) and (ii) above, \(X_{\mathcal {X} }\) does not touch \(\check{X}\), and \(\overline{\check{X} \cup [\cup _{(B,Y)\in \mathcal {X} }B^{\square}]}=U\). The sum over \(X\) in (3.59) has been incorporated into the sum over \((\check{X},\mathcal {X} )\).
Conclusion. We now specialise to the setting of Proposition 3.11. Thus we take \(J\) as in (3.45) and (3.46), and \(\tilde{I}(B)= e^{-(u+V)_{j+1}(B)}\) with the exponent as defined in (3.48), and \(K_{j+1}(U)\) as defined in (3.49). The arguments above show that
What remains is to prove the claims regarding factorisation and automorphism invariance. For factorisation, note that \(\sum _{\mathcal {G} (U_{1}\cup U_{2})} = \sum _{ \mathcal {G} (U_{1})}\sum _{\mathcal {G} (U_{2})}\) for \(U_{1},U_{2}\in \mathcal {P} _{j+1}\) that do not touch. Moreover, since \(U_{1}\) and \(U_{2}\) are separated by a distance at least \(L^{j+1}>\frac{1}{2}L^{j+1}+2^{d+1}L^{j}\), the expectations in the definition of \(K_{j+1}(U_{1}\cup U_{2})\) factor. Here we have used our standing assumption that \(L>2^{d+2}\), that \(J(B,Y)=0\) if \(Y\notin \mathcal {S} _{j}\), and that the range of \(C_{j+1}\) is \(\frac{1}{2}L^{j+1}\). Automorphism invariance follows from the formula for \(K_{j+1}\) and the properties of \(\mathbb{E}_{C_{j+1}}\theta \) discussed in Sect. 3.3. □
Proposition 3.11 implies in particular that if \(K_{j}\) has the factorisation property (3.19), then we can identify \((K_{j+1}(X))_{X\in \mathcal {P} _{j+1}(\Lambda _{N})}\) with its restriction to connected polymers \((K_{j+1}(X))_{X\in \mathcal {C} _{j+1}(\Lambda _{N})}\). Moreover, if the \(K_{j}\) are also automorphism invariant, then \(K_{j+1}\in \mathcal {K} ^{\varnothing }_{j+1}(\Lambda _{N})\).
By construction and the consistency of the covariances \(C_{j}\) with \(j< N\) for different values of \(N\), the maps defined for different \(\Lambda _{N}\) are also consistent in the following sense:
Proposition 3.12
For \(j+1< N\) and \(U \in \mathcal {P} _{j+1}(\Lambda _{N})\), \(V_{j+1}(U)\) and \(K_{j+1}(U)\) above depend on \((V_{j},K_{j})\) only through \(V_{j}(X)\), \(K_{j}(X)\) with \(X \in \mathcal {P} _{j}(U^{\square})\). Moreover, for \(U \in \mathcal {P} _{j+1}(\Lambda _{N}) \cap \mathcal {P} _{j+1}(\Lambda _{M})\) with the natural local identification of \(\Lambda _{N}\) and \(\Lambda _{M}\), the map \((V_{j},K_{j}) \mapsto (V_{j+1}(U),K_{j+1}(U))\) is independent of \(N\) and \(M\).
Temporarily indicating the \(N\)-dependence of \(\Phi _{j+1}=\Phi _{j+1,N}\) explicitly, consistency implies the existence of an infinite volume limit \(\Phi _{j+1,\infty}=\lim _{N\to \infty} \Phi _{j+1,N}\) defined for arguments \(V_{j} \in \mathcal {V} ^{\varnothing }\) and \(K_{j} =(K_{j}(X))_{X\in \mathcal {C} _{j}(\mathbb{Z}^{d})} \in \mathcal {K} ^{\varnothing }_{j}(\mathbb{Z}^{d})\). Explicitly, if we write \(\Phi _{j+1,N}(V_{j},K_{j})= (V^{N}_{j+1},K^{N}_{j+1})\) and omit the \(N\) for the infinite volume map, \(K_{j+1}(U) = \lim _{N\to \infty} K^{N}_{j+1}(U)\), and similarly for \(V_{j+1}\). The limits exist as the sequences are constant after finitely many terms. This infinite volume limit does not carry the full information from the \(\Phi _{j+1,N}\) because terms indexed by polymers that wrap around the torus are lost, but it does carry complete information about small sets at all scales and thus about the flow of \(V_{j}\). As for the finite-volume maps, the infinite volume limit carries a mild dependence on \(m^{2}\). We typically omit this from the notation.
3.8 Estimates for the renormalisation group map
The renormalisation group map \(\Phi _{j+1}= \Phi _{j+1,N}\) is a function of \((V,K)\in \mathcal {V} ^{\varnothing }\oplus \mathcal {K} ^{\varnothing }_{j}(\Lambda _{N})\). The size of \(V\) and \(K\) will be measured in the norms
where \(A>1\) is a parameter that will be chosen sufficiently large. The space of bulk coupling constants \(\mathcal {V} ^{\varnothing }\cong \mathbb{C}^{4}\) has finite dimension. The space for polymer activities \(\mathcal {K} ^{\varnothing }_{j}(\Lambda _{N})\) is also finite-dimensional for \(N<\infty \) since an element \(K_{j} \in \mathcal {K} ^{\varnothing }_{j}(\Lambda _{N})\) is a finite collection of elements \(K_{j}(X)\) of the finite-dimensional Grassmann algebra \(\mathcal {N} (\Lambda _{N})\). Thus \(\mathcal {V} ^{\varnothing }\oplus \mathcal {K} ^{ \varnothing }_{j}(\Lambda _{N})\) is a finite-dimensional complex normed vector space with the above norms, and therefore a Banach space.
Theorem 3.13
Let \(d \geqslant 3\), \(L \geqslant L_{0}(d)\), and \(A \geqslant A_{0}(L,d)\). Assume that \(u_{j}=0\). There exists \(\varepsilon = \varepsilon (L,A)>0\) such that if \(j+1< N\) and \(\|V_{j}\|_{j} + \|K_{j}\|_{j} \leqslant \varepsilon \) then
where \(\eta =\eta (d)\) and \(\nu =\nu (d)\) are positive geometric constants. The maps \(\Phi _{j+1}\) are entire in \((V_{j},K_{j})\) and hence all derivatives of any order are uniformly bounded on \(\|V_{j}\|_{j}+\|K_{j}\|_{j} \leqslant \varepsilon \). Moreover, the maps \(\Phi _{j+1}\) are continuous in \(m^{2} \geqslant 0\).
The last renormalisation group map \(\Phi _{N}\) satisfies the same bound with \(L^{-(\frac{d-2}{2} \wedge 1)}\) replaced by 1.
Theorem 3.13 is the analogue of [31, 32] for the four-dimensional weakly self-avoiding walk, but much simpler since (i) we are only working with fermionic variables, and (ii) we are above the lower critical dimension (two for our model). The factors \(L^{d}\) and \(A^{\nu}\) in the error bounds are harmless. On the other hand, it is essential that \(O(L^{-(\frac{d-2}{2} \wedge 1)}+A^{-\eta}) < 1\) for \(L\) and \(A\) large: this estimate establishes that \(K\) is irrelevant (contracting) in renormalisation group terminology.
The remainder of this subsection proves Theorem 3.13. Readers not familiar with the use of the renormalisation group might want to skip the somewhat technical proof on a first reading and proceed to Sects. 3.9 and 4 to get an idea of how these estimates are used.
Throughout the rest of Sect. 3.8the hypotheses of Theorem 3.13will be assumed to hold.
The substantive claims of Theorem 3.13 are the estimates (3.66) and (3.67): these quickly yield the claims regarding derivatives by a standard Cauchy estimate, as we now explain. Recall that given two Banach spaces \(X\) and \(Y\) and a domain \(D\subset \mathbb{C}\) we say that a function \(g\colon D\rightarrow X\) is analytic if it satisfies the Cauchy-Riemann equation \(\partial _{\bar{z}} g=0\). For an open set \(O\subseteq X\), we then say that a function \(F\colon O \rightarrow Y\) is analytic if \(F\circ g\) is analytic for every analytic function \(g\colon D\rightarrow X\). After (possibly) adding some additional coordinates to ensure all necessary monomials are in the domain, the maps \((V_{j},K_{j}) \mapsto (V_{j+1},K_{j+1})\) are multivariate polynomials, and the norm estimates (3.66) and (3.67) extend to this larger space. Being multivariate polynomials, the \(\Phi _{j+1}\) are analytic functions.
We use analyticity and the Cauchy integral formula to extract derivatives. Namely, if \((V, K)\) and \((\dot{V}^{(i)}, \dot{K}^{(i)})_{i=1}^{n}\) are collections of polymer coordinates at scale \(j\) satisfying \(\|V\|_{j}+\|K\|_{j}\leqslant \varepsilon /2\) and \(\|\dot{V}^{(i)}\|_{j}+\|\dot{K}^{(i)}\|_{j} \leqslant 1\), then
where the \(n\)-tuple of contours are circles around 0 with radius \(\varepsilon /(2 n)\). The statement of Theorem 3.13 regarding boundedness of derivatives follows.
The asserted continuity in \(m^{2} \geqslant 0\) follows from the explicit formulas for \((V_{j+1},K_{j+1})\), that \(\operatorname{Loc}\) is linear, and that the covariances \(C_{j}\) are continuous in \(m^{2} \geqslant 0\).
3.8.1 Coupling constants
We begin with the bound (3.66) for \(V_{j+1}\). The first term on the right-hand side in the definition (3.48) of \(u_{j+1}+V_{j+1}\) produces the expectation term in (3.66). For \(B\in \mathcal {B} _{j}\), the remainder in (3.48) is bounded as follows:
where we have used that the number of small sets containing a fixed block is \(O(1)\) in the second step, and (3.39) in the third. Since each block in \(\mathcal {B} _{j+1}\) contains \(L^{d}\) blocks in \(\mathcal {B} _{j}\), by using (3.34) to bound the expectation and change of scale in the norm, the first claim (3.66) follows.
The remainder of Sect. 3.8 establishes the bound (3.67) for \(K_{j+1}\). The following basic observations will be used repeatedly. Note that by (3.34) the main term contributing to \(u_{j+1}|B|+V_{j+1}(B)\) is bounded by, for \(B\in \mathcal {B} _{j}\),
By combining this with (3.69) we have that, for \(B\in \mathcal {B} _{j}\),
3.8.2 Preparation for bound of the non-perturbative coordinate
To derive (3.67), we first separate from \(K_{j+1}(U)\) a leading contribution. This contribution is:
Note that while the first sum on the right-hand side is over connected polymers, the second is over all polymers. This expression includes the contributions to \(K_{j+1}\) explicitly linear in \(K_{j}\), and all other terms in the definition of \(K_{j+1}\) are higher order, see Sect. 3.8.5 below.
We may divide each of the sums on the right-hand side in (3.72) into the contributions from small sets \(X \in \mathcal {S} _{j}\) and large sets \(X \in \mathcal {P} _{j}\setminus \mathcal {S} _{j}\). We recall that small sets are, by definition, connected. These restricted sums will be denoted by \(\mathcal {L} _{j+1, \mathcal {S} }(U)\) and \(\mathcal {L} _{j+1, \mathcal {P} \setminus \mathcal {S} }(U)\) respectively:
Large sets are easier to handle because they lose combinatorial entropy under change of scale (reblocking), i.e., \(|\mathcal {B} _{j}(X)|\) will be significantly larger than \(|\mathcal {B} _{j+1}(\bar{X})|\). In renormalisation group terminology, they are irrelevant. Small sets, on the other hand, require careful treatment.
3.8.3 Small sets
The main estimate on small sets is summarised as follows.
Proposition 3.14
The contribution \(\mathcal {L} _{j+1,\mathcal {S} }\) to (3.72) satisfies
In order to bound
we consider the terms \(X \in \mathcal {S} _{j} \setminus \mathcal {B} _{j}\) and \(X\in \mathcal {B} _{j}\) in the outer sum separately. By the definition of \(J\) in (3.46), for any \(X \in \mathcal {S} _{j} \setminus \mathcal {B} _{j}\),
where the final equality follows from Proposition 3.8 (iii). Thus the contribution to \(\mathcal {L} _{j+1, \mathcal {S} }\) from \(X \in \mathcal {S} _{j}\setminus \mathcal {B} _{j}\) is
The contribution to \(\mathcal {L} _{j+1, \mathcal {S} }\) from \(X= B \in \mathcal {B} _{j}\) is
We now give a series of estimates, bounding the right-hand sides of (3.77) and (3.78) and the term outside the expectation in (3.75), and then assemble them into a proof for Proposition 3.14.
Lemma 3.15
For any \(U \in \mathcal {C} _{j+1}\),
Proof
Note that \(\bar{X} \in \mathcal {S} _{j+1}\) if \(X \in \mathcal {S} _{j}\), so it suffices to prove the lemma for \(U\in \mathcal {S} _{j+1}\). Now for any \(U \in \mathcal {S} _{j+1}\), since there are \(O(L^{d})\) small sets \(X\in \mathcal {S} _{j}\) such that \(\bar{X}=U\) we get
where we have used the contraction estimate (3.33) for the expectation in the second step and the contraction estimate (3.40) for \(\operatorname{Loc}_{X}\) in the third step. □
Lemma 3.16
For \(B \in \mathcal {B} _{j}\),
Proof
By the definition of \((u+V)_{j+1}\) in (3.48) we have
By the product property (3.26), if for some \(V\) and some \(k\) we have \(\|V(B)\|_{T_{k}(\ell _{k})} \leqslant 1\), then
Recall that \(\mathbb{E}_{C_{j+1}}\theta \) is contractive as a map from \(T_{j}(\ell _{j})\) to \(T_{j+1}(\ell _{j+1})\) by (3.34). Applying these estimates to the \(T_{j+1}(\ell _{j+1})\) norm of (3.82) and using (3.71) gives the bound (3.81). □
Lemma 3.17
For \(X \in \mathcal {P} _{j}\),
Proof
Using that \(\mathbb{E}_{C_{j+1}}\) satisfies the contraction estimate (3.32), it suffices to show
By the product property (3.26) it suffices to prove this estimate for a single block. In this case,
by the product property (3.26) of the norms and (3.31). Using \(2\ell _{j+1} \leqslant \ell _{j}\) and (3.30) for the first term and (3.71) for the second term bounds the right-hand side by \(O(\|V_{j}\|_{j} + \|K_{j}\|_{j})\) as needed. □
We need one further general estimate.
Lemma 3.18
If \(\|K_{j}\|_{j} + \|V_{j}\|_{j} \leqslant \varepsilon =\varepsilon (d,L)\) is sufficiently small, then if \(\bar{X} = U \in \mathcal {P} _{j+1}\),
Proof
By the product property (3.26) and (3.71) to bound \(V_{j+1}\) and \(u_{j+1}\),
and \(|\mathcal {B} _{j}(U)|\) is at most \(L^{d}|\mathcal {B} _{j+1}(U)|\leqslant L^{d}| \mathcal {B} _{j}(X)|\). The claim follows provided \((1+{ O(}\varepsilon ))^{L^{d}}\leqslant 2\). □
Proof of Proposition 3.14
To estimate the summands of \(\mathcal {L} _{j+1, \mathcal {S} }(U)\), we use the product property of the \(\|\cdot \|_{T_{j+1}(\ell )}\) norm to combine Lemma 3.18 with Lemma 3.15, Lemma 3.16 for \(X\in \mathcal {B} _{j}\), and with Lemma 3.17 for \(X\in \mathcal {S} _{j}\backslash \mathcal {B} _{j}\). For the sum of the terms \((\delta I)^{X}\) we use that \((1+\|V_{j}\|_{j}^{2} + \|K_{j}\|_{j}^{2})^{L^{d}}\leqslant 2\) provided \(\varepsilon =\varepsilon (L)\) is small enough. Altogether, we obtain
which proves the lemma. □
3.8.4 Large sets
Next we consider the contribution to (3.72) from the terms \(X \notin \mathcal {S} _{j}\) in the sums, i.e., \(\mathcal {L} _{j+1,\mathcal {P} \setminus \mathcal {S} }\). The main estimate of this section is summarised in the following proposition.
Proposition 3.19
The contribution \(\mathcal {L} _{j+1,\mathcal {P} \setminus \mathcal {S} }\) to (3.72) satisfies
We begin by recording a combinatorial fact, see [28, Lemmas 6.15 and 6.16] or [32, Lemma C.3] for details on its proof. For the statement, recall that if \(X\in \mathcal {P} _{j}\), then its closure \(\bar{X}\in \mathcal {P} _{j+1}\) denotes the smallest next-scale polymer containing \(X\).
Lemma 3.20
Let \(L \geqslant 2^{d}+1\). There is a geometric constant \(\eta =\eta (d)>0\) depending only on \(d\) such that for all \(X \in \mathcal {C} _{j} \setminus \mathcal {S} _{j}\),
Moreover, for all \(X \in \mathcal {P} _{j}\), \(|\mathcal {B} _{j}(X)| \geqslant |\mathcal {B} _{j+1}(X)|\) and
We also record an application of this estimate to sums indexed by large polymers which will be used in this and in the next section. By (3.91), if \(A=A(L)\) is large enough,
as the set of \(X\in \mathcal {P} _{j}\) with \(\bar{X}=U\) has size at most \(2^{L^{d}|\mathcal {B} _{j+1}(U)|}\). Similarly, by (3.92), if \(\alpha \geqslant A^{(1+\eta )2^{d +1}}\),
Proof of Proposition 3.19
From (3.72) and (3.73), recall that (as \(J(B,X)=0\) for large \(X\))
We first consider the case \(U\in \mathcal {C} _{j+1}\setminus \mathcal {S} _{j+1}\), and proceed as follows: for \(\|K_{j}\|_{j} + \|V_{j}\|_{j} \leqslant \varepsilon \) with \(\varepsilon \) sufficiently small, by Lemma 3.18 the \(j+1\) norm of the \(K\) contribution to (3.95) is bounded by
By the definition of \(\|K_{j}\|_{j}\) and noting that \((|\mathcal {B} _{j}(X)|-2^{d})_{+}=| \mathcal {B} _{j}(X)|-2^{d}\) since \(X\notin \mathcal {S} _{j}\),
where we have also used the contraction estimates (3.33) and (3.30). Inserting this bound into (3.96) and using (3.93) gives that the \(K\) contribution to (3.95) is bounded by
This is the desired bound for the first term in (3.95).
To bound the \(j+1\) norm of the \(\delta I\) contribution to (3.95), Lemmas 3.17 and 3.18 and the product property yield (provided \(\varepsilon \) is sufficiently small depending on \(L\))
Since \(U \in \mathcal {C} _{j+1} \setminus \mathcal {S} _{j+1}\) and \(\bar{X}=U\), each \(X\) in the last sum must have \(|\mathcal {B} _{j}(X)|\geqslant 2^{d}+1\). If \(\|V_{j}\|_{j} + \|K_{j}\|_{j}<\varepsilon \) and \(\varepsilon \) is sufficiently small (depending on \(A\)), then the quantity in brackets is less than \(1/A^{2+2(1+\eta )2^{d+1}}\). By the elementary inequality \((c^{2})^{n-2}\leqslant c^{n}\) for \(c\in (0,1)\), \(n>4\) and using that \(|\mathcal {B} _{j}(X)|\geqslant 2^{d}+1>4\) for each summand, we obtain the upper bound
Using (3.94), it follows that the \(\delta I\) contribution to (3.95) is bounded by
for \(A\) sufficiently large. We have now completed the bound on (3.72) provided \(U\in \mathcal {C}_{j+1}\backslash \mathcal {S}_{j+1}\).
The argument is similar if \(U \in \mathcal {S} _{j+1}\). In this case the prefactor \(A^{|\mathcal {B} _{j+1}(U)|-2^{d}}\) gets replaced by 1 in (3.96) and (3.99). For the \(K\) contribution, in place of (3.98) we obtain, since \(1+ 2^{d}\leqslant |\mathcal {B} _{j}(X)|\leqslant L^{d} | \mathcal {B} _{j+1}(U)|\leqslant (2L)^{d}\) and the number of summands in this case is at most \(2^{(2L)^{d}}\),
for \(A\) large enough depending on \(L\) and \(d\). For the \(\delta I\) contribution, in place of (3.99) we have
since each summand on the left-hand side has \(|\mathcal {B} _{j}(X)|\geqslant 2\).
Thus for \(A=A(L,d)\) sufficiently large and \(\varepsilon =\varepsilon (A,L)\) sufficiently small, the expression (3.95) is bounded in the \(T_{j+1}(\ell _{j+1})\) norm by \(O(A^{-\eta}\|K_{j}\|_{j}+A^{\nu}[\|V_{j}\|_{j} + \|K_{j}\|_{j}]^{2})\) in all cases. □
3.8.5 Non-linear part
Finally, we consider the non-linear contribution \(K_{j+1}-\mathcal {L} _{j+1}\). To conclude the proof of (3.67), and hence the proof of Theorem 3.13, we prove the following estimate.
Proposition 3.21
Before diving into the proof, let us review the terms which remain to be estimated. Recall the definition of \(K_{j+1}(U)\) from (3.49) and the leading part \(\mathcal {L} _{j+1}(U)\) from (3.72). Write \(|\mathcal {X} |\) for the number of pairs in \(\mathcal {X} \). With respect to the indexing of summands for \(K_{j+1}(U)\), the leading part \(\mathcal {L} _{j+1}(U)\) results from the terms with \(|\mathcal {X} |=0\) and \(\check{X}=X\) by only keeping the terms in the formula for \(\check{K}(X)\) with either a single factor \(\theta K_{j}(X)\) when \(X\in \mathcal {C} _{j}\), a single factor \((\delta I)^{X}\) when \(X\in \mathcal {P} _{j}\), or a single factor \(\sum _{B} \theta J(B,X)\). It follows that we can write
where
and
when the subsets \(\mathcal {G} _{i}(U)\subset \mathcal {G} (U)\) are defined as follows. The set \(\mathcal {G} _{1}(U)\) is defined by imposing the conditions \(|\mathcal {X} |=1\) and \(\check{X} = \varnothing \). The set \(\mathcal {G} _{2}(U)\) is defined to consist of \((\mathcal {X} , \check{X})\) such that \(X_{\mathcal {X} }\cup \check{X}\) has at least two components. In particular, if \(\mathcal {X} =\varnothing \), \(\check{X}\) has least two components and if \(\check{X}=\varnothing \) then \(|\mathcal {X} |\geqslant 2\). Finally, \(\mathcal {G} _{3}(U)\) is defined by the conditions \(|\mathcal {X} |=0\) and \(\check{X}\in \mathcal {C} _{j}\).
The next lemma clearly implies Proposition 3.21.
Lemma 3.22
For \(i\in \{1, 2, 3\}\),
Proof of Lemma 3.22 for \(i=1\)
We begin by bounding \(\mathcal {R} ^{1}\). This bound exploits that \(\sum _{X} J(B,X)=0\) for every \(B\in \mathcal {B} _{j}\), see (3.45)–(3.46) and (3.47). As \(\mathcal {X} \) is a single pair \(\{ (B,X)\}\), \(X_{\mathcal {X} }=X\). Since \(\check{X}=\varnothing \) we can write
Since \(\sum _{X_{\mathcal {X} }} J(B,X_{\mathcal {X} })=0\) for \(B\in \mathcal {B} _{j}\), we can rewrite
Since \(X_{\mathcal {X} } \in \mathcal {S} _{j}\) we have \(\|1-e^{(u+V)_{j+1}(X_{\mathcal {X} })}\|_{T_{j+1}(\ell _{j+1})} = O(L^{d}(\|V_{j}\|_{j}+\|K_{j}\|_{j}))\) by (3.71). Moreover, (3.39) implies \(\|J(B, X)\|_{T_{j}(\ell _{j})} =O(\|K_{j}\|_{j})\), so \(\|\mathbb{E}_{C_{j+1}}\theta J(B, X)\|_{T_{j+1}(\ell _{j+1})} =O(\|K_{j} \|_{j})\) since \(\mathbb{E}_{C_{j+1}}\theta \) is a contraction. Finally, exactly as in the proof of Lemma 3.18, \(\|e^{u_{j+1}(U)}e^{-(u+V)_{j+1}(U\setminus X_{ \mathcal {X} })}\|_{T_{j+1}(\ell _{j+1})}\leqslant (1+O( \varepsilon ))^{|\mathcal {B} _{j}(U)|}\leqslant 2\) for \(\varepsilon =\varepsilon (L)\) small enough, since \(U\) is the closure of \(B^{\square}\). As there are \(O(L^{d})\) summands in (3.111) we have shown
Since \(A^{(|\mathcal {B} _{j+1}(U)|-2^{d})_{+}}=1\) for any contributing \(U\) (as \(U\) is the closure of \(B^{\square}\) for some block \(B\)), this concludes the desired bound on \(\mathcal {R} ^{1}(U)\). □
Before proceeding to the proof of Lemma 3.22 for \(i=2\), we first provide estimates on the norms of \(\check{K}(\check{X})\) and \(\check{K}(\check{X})-(\delta I)^{\check{X}}\).
Lemma 3.23
If \(\|V_{j}\|_{j}+\|K_{j}\|_{j}\leqslant \varepsilon \) and \(\varepsilon =\varepsilon (A, L)\) is sufficiently small, then
The proof of these estimates is postponed until the conclusion of the main argument.
Proof of Lemma 3.22 for \(i=2\)
If \(\|V_{j}\|_{j} + \|K_{j}\|_{j}\) is small enough, arguing as in (3.87) implies \(\|e^{u_{j+1}(U)}e^{-(u+V)_{j+1}(U\setminus \check{X} \cup X_{ \mathcal {X} })}\|_{T_{j+1}(\ell _{j+1})}\leqslant 2^{| \mathcal {B} _{j}(U)|}\), and by (3.39), \(\|J(B, X)\|_{T_{j}(\ell _{j})} =O(\|K_{j}\|_{j})\). Thus using that \(\mathbb{E}_{C_{j+1}}\) contracts from \(T_{j+1}(\ell _{j+1}\sqcup \ell _{j+1})\) into \(T_{j+1}(\ell _{j+1})\) we obtain
For brevity let us write \(b\) for the factors \(O(\|V_{j}\|_{j}+\|K_{j}\|_{j})\) above. By (3.115) and Lemma 3.23, it suffices to show
Indeed, (3.115) is bounded by \(O(A^{-|\mathcal {B} _{j+1}(U)|}\|K_{j}\|_{j}/b)\) times this quantity. The small \(\|K_{j}\|_{j}/b\) is due to the fact that if \(|\mathcal {X} |\geqslant 1\) there is a factor \(\|K_{j}\|_{j}\) in (3.115) and if \(|\mathcal {X} |=0\) then (3.114) provides such a factor in place of \(b\). Hence (3.116) gives
To verify (3.116), first note that since \(|\mathcal {B} _{j}(U)|\leqslant L^{d}| \mathcal {B} _{j+1}(U)|\), for any \(c>0\) the prefactor can be bounded by
Taking \(c>1\), the product of the first two terms on the last right-hand side is less than 1 for \(A\) sufficiently large. It thus suffices to prove that for some \(c>1\)
At this point we appeal to [28, proof of Lemma 6.17]; this result estimates the same sum but over \(\mathcal {G} (U)\) instead of \(\mathcal {G} _{2}(U)\). However, following exactly the same proof as in [28] but using that the sum is over \(\mathcal {G} _{2}(U)\), the supremum over \(n\geqslant 1\) in [28, (6.85)] becomes a supremum over \(n\geqslant 2\) since \(|\mathcal {X} | + |\operatorname{Comp}(\check{X})| \geqslant 2\). This yields that there exists \(c>1\) such that if \(A=A(L,d)\) is large enough, then there is an \(m\) such that for all \(U\in \mathcal {P} _{j+1}\),
which is \(A^{\nu}O(b^{2})\) as needed. □
Proof of Lemma 3.23
For notational convenience, for \(Y\in \mathcal {C}_{j}\) let
We first establish that the claimed bounds follow from the definition of \(\check{K}(X)\) in (3.50) if we show, for \(Y\in \mathcal {C} _{j}\),
Indeed, though \(\check{K}(\check{X}) - (\delta I)^{\check{X}}\) does not factor over components \(X\) of \(\check{X}\), it can be written as a sum of \(|\operatorname{Comp}(\check{X})|\) terms, each of which is a product over the components \(X\) of \(\check{X}\). That is, we use the formula \((a+b)^{n}-a^{n}=\sum _{k=0}^{n-1}a^{k}b(a+b)^{n-k-1}\) with \(a=(\delta I)^{X}\) and \(b=\check{K}(X)-(\delta I)^{X}\). Thus each summand contains one factor \(\check{K}(X)-(\delta I)^{X}\) and the rest of the factors are either \(\check{K}(X)\) or \((\delta I)^{X}\). The estimates (3.113)-(3.114) then follow by using (3.122)-(3.123) and Lemma 3.17.
To establish (3.122)–(3.123) we apply the triangle inequality. Since \(J(B,Y)=0\) if \(Y\notin \mathcal {S} _{j}\),
where we have used \(\|J(B, X)\|_{T_{j}(\ell _{j})}=O(\|K_{j}\|_{j})\), that \(\theta \) contracts from \(T_{j}(\ell _{j})\) into \(T_{j+1}(\ell _{j+1})\) and that \(|\mathcal {B} _{j}(Y)|\leqslant 2^{d}\). This shows the \(J\) contributions to (3.122) and (3.123) satisfy the requisite bounds. For the other contributions, note that by (3.85), component factorisation of \(K_{j}\), and the contraction property of the norms and \(\theta \), for \(B\in \mathcal {B} _{j}\) and \(Z\in \mathcal {P} _{j}\) we have
We now impose the condition that \(\varepsilon \leqslant A^{-2^{d}}\) and that \(O(\varepsilon ) \leqslant A^{-1}\) in the implicit bound below. Plugging the previous bounds into the expression for \(\tilde{K}(Y)\) we have
Since \(\|(\delta I)^{Y}\|_{T_{j+1}(\ell _{j+1})} \leqslant [O(\|V_{j}\|_{j}+ \|K_{j}\|_{j})]^{|\mathcal {B} _{j}(Y)|} \leqslant A O(\|V_{j} \|_{j}+\|K_{j}\|_{j}) A^{-|\mathcal {B} _{j}(Y)|}\) if \(O(\varepsilon )\leqslant A^{-1}\), by the triangle inequality we also have
Together with (3.124) this proves the lemma. □
Proof of Lemma 3.22 for \(i=3\)
The bound on \(\mathcal {R} ^{3}(U)\) is similar to that of \(\mathcal {R} ^{2}(U)\) but simpler since only connected \(\check{X}\) are involved. In particular, the analogue of Lemma 3.23 only involves the reasoning leading to (3.127), with the key difference being that now also \(Z\neq \emptyset \). We omit the details. □
3.9 Flow of the renormalisation group
Recall the infinite volume limit of the renormalisation group maps \(\Phi _{j+1,\infty}\) discussed below Proposition 3.12. We equip \(\mathcal {K} ^{\varnothing }_{j}(\mathbb{Z}^{d})\) with the norm \(\|K\|_{j}\) defined by (3.65). This space is now infinite dimensional, but it is clear that it is still complete as a normed space, i.e., a Banach space. Moreover, by the consistency of the finite volume renormalisation group maps (Proposition 3.12), the estimates given in Theorem 3.13 also hold for the infinite volume limit. Next we study the iteration of the renormalisation group maps as a dynamical system. In what follows \(K_{0}=0\) means \(K_{0}(X)=1_{X=\varnothing}\) for \(X\in \mathcal {P} _{j}\).
In the next theorem (and later in the paper) we write \(O_{L}(\cdot )\) to indicate a bound with a constant that may depend on \(L\), but where the constant is uniform in \(j\), i.e., \(f_{j} = O_{L}(L^{-j})\) if there is a \(C=C(L)\) such that \(f_{j}\leqslant C L^{-j}\) for all \(j\).
Theorem 3.24
Let \(d\geqslant 3\), \(L\geqslant L_{0}\), and \(A \geqslant A_{0}(L)\). For \(m^{2} \geqslant 0\) arbitrary and \(b_{0}\) small, there exist \(V_{0}^{c}(m^{2},b_{0})\) and \(\kappa >0\) such that if \((V_{0},K_{0})=(V_{0}^{c}(m^{2},b_{0}),0)\) and \((V_{j+1},K_{j+1})= \Phi _{j+1,\infty ,m^{2}}(V_{j},K_{j})\) is the flow of the infinite volume renormalisation group map then
The components of \(V_{0}^{c}(m^{2},b_{0})\) are continuous and uniformly bounded in \(m^{2} \geqslant 0\) and differentiable in \(b_{0}\) with uniformly bounded derivative.
Proof of Theorem 3.24
The proof is by a version of the stable manifold theorem for smooth dynamical systems. Specifically, we use [28, Theorem 2.16].
To start, we write down the dynamical system corresponding to the renormalisation group map. The definition of \(V_{j+1}\) is (3.48). We start with the contribution to \(V_{j+1}\) arising from the first term
where \(\tilde{u}_{j+1}\) and \(\tilde{V}_{j+1}\) are defined by the right-hand side. These can be computed by the Wick formula (3.10), which gives
with \(\kappa _{j}^{yb} = -C_{j+1}(0)\), \(\kappa _{j}^{ab}=\Delta C_{j+1}(0)\), and \(\kappa _{j}^{zb} = \frac{1}{2d}\Delta C_{j+1}(0)\). Indeed, non-constant contributions only arise from quartic terms, and Wick’s formula gives
Since \(\nabla _{e} C(0) = C(e)-C(0) = \frac{1}{2d}\sum _{e\in \mathcal {E} _{d}} (C(e)-C(0))= \frac{1}{2d}\Delta C(0)\) for all \(e\in \mathcal {E} _{d}\) by symmetry, and since the lattice Laplacian has the representations \(\Delta = -\frac{1}{2} \sum _{e\in \mathcal {E} _{d}} \nabla _{e}\nabla _{-e} = \sum _{e\in \mathcal {E} _{d}} \nabla _{e}\), therefore
Since \(\|V_{j}(B)\|_{T_{j}(\ell _{j})}\) is comparable with \(|z_{j}| + |y_{j}| + L^{2j}|a_{j}| + L^{-(d-2)j} |b_{j}|\), i.e., \(\|V_{j}(B)\|_{T_{j}(\ell _{j})} = O_{L}(|z_{j}| + |y_{j}| + L^{2j}|a_{j}| + L^{-(d-2)j} |b_{j}|) = O_{L}(\|V_{j}(B)\|_{T_{j}(\ell _{j})})\), it is natural to define the rescaled variables and coefficients \(\hat{z}_{j}=z_{j}\), \(\hat{y}_{j}=y_{j}\), \(\hat{a}_{j}=L^{2j}a_{j}\), \(\hat{b}_{j}=L^{-(d-2)j}b_{j}\), \(\hat{\kappa}_{j}^{ab}=L^{2+dj}\kappa _{j}^{ab}\), \(\hat{\kappa}_{j}^{yb}=L^{(d-2)j}\kappa _{j}^{yb}\), and \(\hat{\kappa}_{j}^{zb}=L^{(d-2)j}\kappa _{j}^{zb}\). The definition (3.48) of \(V_{j+1}\) then becomes
Here \(\hat{r}_{j}\) is the 4-component vector of real numbers determined by the \(\operatorname{Loc}\) step of the renormalisation group map. Each component is thus a linear function of \(K_{j}\), and by (3.66) of Theorem 3.13 has size \(O(L^{d}\lVert K_{j} \rVert _{j})\). The \(\hat{\kappa}_{j}\) are uniformly bounded in \(j\) by the covariance estimates (3.3).
We now reorganize variables. Set \(v_{j}=(\hat{y}_{j},\hat{z}_{j},\hat{a}_{j})\) and \(w_{j}=(\hat{b}_{j},K_{j})\) and use \(\|\cdot \|_{j}\) for the norm given by maximum of the (norm of the) respective components. The index \(j\) does not play a role for \(\|v_{j}\|_{j}\), but it does for \(\|w_{j}\|_{j}\). By the computation above and Theorem 3.13 the infinite volume renormalisation group map can be written in the block diagonal form
In this formula, \(E\) comes from the first terms on the right-hand sides of the \(\hat{z}\), \(\hat{y}\), and \(\hat{a}\) equations in (3.134) and (3.135), \(B_{j}\) represents the \(\hat{\kappa}^{xb}_{j}\) and the \(\hat{r}^{x}_{j}\) terms with \(x=z,y,a\), and \(D_{j}\) represents the first term in the \(\hat{b}\) equation in (3.135) and the linearisation of \((0, K_{j})\mapsto K_{j+1}\). Finally, \(g_{j+1}(v_{j}, w_{j})\) is then the non-linear remainder of \(K_{j+1}\) after the linearisation is removed. It follows from these identifications that \(g_{j}(0,0)=0\) and \(Dg_{j}(0,0)=0\). Moreover \(g_{j}\) is analytic in its arguments, so that all structural hypotheses required to apply [28, Theorem 2.16] hold.
As for the requisite norm estimates, since it is a \(3\times 3\) triangular matrix with non-zero diagonal entries, \(E\) is invertible with bounded inverse \(E^{-1}\). As indicated above, \(\|\hat{r}_{j}\|_{j+1}=O(L^{d}\|K_{j}\|_{j})\), so \(\|B_{j}\|_{j\to j+1}\) is bounded. Finally, the derivative estimates on the renormalisation group map from Theorem 3.13 imply the following norm bounds on \(D_{j}\):
with the latter inequality holding provided \(A\) is large enough.
For every \(m^{2}\geqslant 0\), \(b_{0}\) sufficiently small, and \(K_{0}=0\), [28, Theorem 2.16] now implies that we can find an initial 3-tuple of coupling constants \(v_{0}^{c}(m^{2},b_{0})\), or equivalently an initial local potential \(V_{0}^{c}(m^{2},b_{0})\), so that for some \(\kappa >0\),
These bounds are not explicit in the statement of [28, Theorem 2.16], but are immediate from its proof (because the proof constructs a solution in a correspondingly weighted sequence space). In particular this bound implies that \(\|K_{j}\|_{j}+\|V_{j}\|_{j}= O_{L}(\|v_{j}\|_{j}+ \|w_{j}\|_{j})=O_{L}(b_{0}L^{- \kappa j})\).
Smoothness of the renormalisation group map implies that \(V_{0}^{c}(m^{2},b_{0})\) is smooth in \(b_{0}\). To see that \(V_{0}^{c}(m^{2},b_{0})\) is also continuous in \(m^{2} \geqslant 0\), one can for example regard \(v_{j}\) and \(w_{j}\) as bounded continuous functions of \(m^{2}\), i.e., consider \(v_{j} \in C_{b}([0,\infty ),\mathbb{R}^{3})\) and \(w_{j}\in C_{b}([0,\infty ), \mathbb{R}\times \mathcal {K} ^{ \varnothing }_{j}(\mathbb{Z}^{d}))\). Since all the estimates above are uniform in \(m^{2} \geqslant 0\), the previous argument applies in these spaces and shows that the solution is continuous in \(m^{2}\). □
Remark 3.25
Note that while Theorem 3.13 assumes that \(u_{j}=0\) and produces \(u_{j+1}\), it is trivial to extend the statement to \(u_{j}\neq 0\) by simply adding \(u_{j}\) to the \(u_{j+1}\) produced for \(u_{j}=0\).
By consistency, the finite volume renormalisation group flow for \(V_{j}\) agrees with the infinite volume renormalisation group flow up to scale \(j< N\) provided both have the same initial condition. As a result we obtain the following corollary by iterating the recursion (3.67) for the \(K\)-coordinate using the a priori knowledge that \(\|V_{j}\|_{j}=O_{L}(b_{0}L^{-\kappa j})\) due to Theorem 3.24.
Corollary 3.26
Under the same assumptions as in Theorem 3.24, the same estimates hold for the finite volume renormalisation group flow for all \(j \leqslant N\), and the \(V_{j}\) and \(u_{j}\) produced by the finite volume renormalisation group flow are the same as those for the infinite volume flow when \(j< N\).
From this it follows that if \(e^{-u_{N}|\Lambda _{N}|}\) denotes the total prefactor accumulated in the renormalisation group flow up to scale \(N\), then \(u_{N}\) is uniformly bounded in \(N\) and \(m^{2}\) as \(m^{2}\downarrow 0\) if we begin with \(V_{0}\) as in Theorem 3.24. Indeed, up to scale \(N-1\) this follows from the bounds (3.129) and (3.71). In passing from the scale \(N-1\) to \(N\), the renormalisation group step is \(\Lambda _{N}\)-dependent, but is nevertheless uniformly bounded by the last statement of Theorem 3.13.
4 Computation of the susceptibility
In the remainder of the paper, we will use the notation (with \(\Lambda = \Lambda _{N}\))
and assume that \((V_{j},K_{j})_{j=0,\dots ,N}\) is a renormalisation group flow, i.e., \((V_{j+1}, K_{j+1})=\Phi _{j+1}(V_{j},K_{j})\).
In this section we express the susceptibility in terms of the dynamical system generated by the (bulk) renormalisation group flow. First recall that \(Z_{0}=e^{-V_{0}(\Lambda )}\) and that
where \(\Delta \) is the Laplacian on \(\Lambda _{N}\), \(t_{N} = m^{-2}- O(L^{2N})\) is a constant, and the matrix \(Q_{N}\) is the orthogonal projection onto constants (i.e., all entries equal \(1/|\Lambda _{N}|\)). Using (3.11) and (3.13), with \(u_{N}\) as in (3.18), we then set
where \(\mathbb{E}_{t_{N}Q_{N}}\theta \) is the fermionic Gaussian convolution with covariance \(t_{N}Q_{N}\) defined in Sect. 3.1. Thus \(\tilde{Z}_{N,N}\) is still a function of \(\psi \), \(\bar{\psi}\), i.e., an element of \(\mathcal {N} (\Lambda )\). Note that \(\mathbb{E}_{C}[Z_{0}]\) is the constant term of \(Z_{N,N}\), i.e., obtained from \(Z_{N,N}\) by formally setting \(\bar{\psi}\) and \(\psi \) to 0. The following technical device of restricting to constant fields \(\psi \), \(\bar{\psi}\) will be useful for extracting information. By restriction to constant \(\psi \), \(\bar{\psi}\) we mean applying the homomorphism from \(\mathcal {N} (\Lambda )\) onto itself that acts on the generators by \(\psi _{x} \mapsto \frac{1}{|\Lambda |}\sum _{x\in \Lambda} \psi _{x}\) and likewise for the \(\bar{\psi}_{x}\). The result is an element in the subalgebra of \(\mathcal {N} (\Lambda )\) generated by \(\frac{1}{|\Lambda |}\sum _{x\in \Lambda} \psi _{x}\) and \(\frac{1}{|\Lambda |}\sum _{x\in \Lambda} \bar{\psi}_{x}\); we will simply denote these generators by \(\psi \) and \(\bar{\psi}\) when no confusion can arise. To explain the notation in the next proposition, note that a general even element of this subalgebra can be written as \(F^{0}+F^{2}\psi \bar{\psi}\) for some constants \(F^{0}\), \(F^{2}\), c.f. (3.21).
Proposition 4.1
Restricted to constant \(\psi \), \(\bar{\psi}\),
where
If \(V_{0}\), \(K_{0}\) are continuous in \(m^{2}\geqslant 0\) and \(b_{0}\) small enough, then so are \(k_{N}^{0}\) and \(k^{2}_{N}\).
Proof
Since the set of \(N\)-polymers \(\mathcal {P} _{N}(\Lambda _{N})\) is \(\{\varnothing , \Lambda _{N}\}\) and \(e^{u_{N}|\Lambda _{N}|}\) is a constant, (3.18) and (4.3) simplify to
We now evaluate the integral over the zero mode with covariance \(t_{N}Q_{N}\). To this end, we restrict \(V_{N}(\Lambda _{N})\) and \(K_{N}(\Lambda _{N})\) to spatially constant \(\psi \), \(\bar{\psi}\) and denote these restrictions by \(\widetilde{V}_{N} (\Lambda _{N})\) and \(\widetilde{K}_{N}(\Lambda _{N})\). Since \(\widetilde{V}_{N}\) and \(\widetilde{K}_{N}\) are even, they are of the form
where the form of \(\widetilde{V}_{N}\) follows from the representation (3.36). Thus restricted to constant \(\psi \) and \(\bar{\psi}\) the integrand in (4.6) is
Therefore applying the fermionic Wick formula \(\mathbb{E}_{t_{N}Q_{N}}\theta \psi \bar{\psi}= -t_{N}|\Lambda _{N}|^{-1} +\psi \bar{\psi}\), we obtain (4.4). The continuity claims for \(k_{N}^{0}\) and \(k_{N}^{2}\) follow as \((V_{j},K_{j})\) is a renormalisation group flow (see below (4.1)) and since the renormalisation group map has this continuity.
The bounds (4.5) follow from the definition of the \(T_{N}(\ell _{N})\) norm. Indeed, since \(k_{0}\) is the constant coefficient of \(K_{N}(\Lambda _{N})\), clearly \(k_{N}^{0}=O(\|K_{N}\|_{N})\). Similarly, setting \(g_{(x,1),(y,-1)} = 1\) for all \(x,y\in \Lambda _{N}\) and \(g_{z}=0\) for all other sequences, we have \(\|g\|_{\Phi _{N}(\ell _{N})} = \ell _{N}^{-2}\) and
where \(\langle \cdot ,\cdot \rangle \) is the pairing from Definition 3.3. □
Proposition 4.2
Using the notation of Proposition 4.1,
Proof
We amend the algebra \(\mathcal {N} (\Lambda _{N})\) by two Grassmann variables \(\sigma \) and \(\bar{\sigma}\) which we view as constant fields \(\sigma _{x}=\sigma \) and \(\bar{\sigma}_{x}=\bar{\sigma}\). We then consider the fermionic cumulant generating function (an element of the Grassmann algebra generated by \(\sigma \) and \(\bar{\sigma}\))
where \(C\) is as in (4.2). By translation invariance
The linear change of generators \(\psi \mapsto \psi +C\sigma \), \(\bar{\psi}\mapsto \bar{\psi}+ C\bar{\sigma}\) yields
where the right-hand side is to be interpreted as applying the doubling homomorphism and then restricting to constant \(\sigma \), \(\bar{\sigma}\). Since \(\sigma \) is constant in \(x\in \Lambda _{N}\), we have \(C\sigma = m^{-2} \sigma \). With (4.3) thus
□
5 The observable renormalisation group flow
Recall that \(\langle \cdot \rangle \) denotes the expectation (4.1), in which we will ultimately choose \(V_{0}=V_{0}^{c}(b_{0},m^{2})\) as in Theorem 3.24. This section sets up and analyses the renormalisation group flow associated to source fields. This will enable the computation of correlation functions (observables) like \(\langle \bar{\psi}_{a}\psi _{b} \rangle \) in Sect. 6. Our strategy is inspired by that used in [14, 75], but with important differences arising due to the presence of a non-trivial zero mode in our setting.
5.1 Observable coupling constants
As in the proofs in Sect. 4, we amend the Grassmann algebra by two source fields. Now, however, the additional fields are not constant in space but rather are localised at two points \(a,b\in \Lambda = \Lambda _{N}\). We distinguish between two cases:
Case (1). For the two point function \(\langle \bar{\psi}_{a}\psi _{b} \rangle \) (which we call ‘Case (1)’), the additional source fields \(\sigma _{a}\) and \(\bar{\sigma}_{b}\) are two additional Grassmann variables that anticommute with each other and the \(\psi \), \(\bar{\psi}\).
Case (2). For the quartic correlation function \(\langle \bar{\psi}_{a}\psi _{a}\bar{\psi}_{b}\psi _{b} \rangle \) (called ‘Case (2)’), we introduce additional Grassmann variables \(\bar{\vartheta}_{x}\), \(\vartheta _{x}\) for \(x\in \{a,b\}\) and the additional source fields \(\sigma _{a}\) and \(\sigma _{b}\) are the commuting variables \(\sigma _{x}=\bar{\vartheta}_{x}\vartheta _{x}\) for \(x\in \{a,b\}\). This explicit \(U(1)\) invariant choice will be convenient when discussing symmetries.
In both cases we relabel the initial potential \(V_{0}\) from Sect. 3 as \(V^{\varnothing }_{0}\) and set \(V_{0} = V^{\varnothing }_{0}+V^{\star }_{0}\) where \(V^{\star }_{0}\) is an observable part to be defined.
Case (1). In this case,
The spatial index of \(V^{\star }_{0}\) signals the local nature of the source fields. More precisely, the evaluation of \(V^{\star }_{0}\) on a set \(X\) is defined to be spatially localised: \(V^{\star }_{0}(X) = - \lambda _{a,0} \sigma _{a} \bar{\psi}_{a} 1_{a \in X} - \lambda _{b,0}\psi _{b} \bar{\sigma}_{b} 1_{b\in X}\). Recalling that \(C=(-\Delta + m^{2})^{-1}\), see (3.1), it follows that
Obtaining (5.2) is just a matter of expanding \(e^{-V_{0}^{\star }(\Lambda )}\), using \(\langle \bar{\psi}_{a} \rangle =\langle \psi _{b} \rangle =0\), and applying the rules of Grassmann calculus. Note the order of \(\partial _{\bar{\sigma}_{b}}\) and \(\partial _{\sigma _{a}}\), which is important to obtain the correct sign. Although (5.2) holds for any constants \(\lambda _{a,0}\), \(\lambda _{b,0}\), it is convenient for us to leave these as variables to be tracked with respect to the renormalisation group flow.
Case (2). Similarly to the previous case, we choose
so that
To distinguish the coupling constants in the two cases, we will sometimes write \(\lambda _{a,0}^{(p)}\) with \(p=1\) or \(p=2\) instead of \(\lambda _{a,0}\), and analogously for the other coupling constants.
5.2 The free observable flow
To orient the reader and motivate the discussion which follows, let us first consider the noninteracting case \(V^{\varnothing }_{0}=0\), in which the microscopic model is explicitly fermionic Gaussian. In this case, one may compute all correlations explicitly by applying the fermionic Wick rule. The same computation can be carried out inductively using the finite range decomposition of the covariance \(C\), and we review this now as it will be the starting point for our analysis of the interacting case.
To begin the discussion, observe that all source fields square to zero, i.e., \(\sigma _{a}^{2} = \bar{\sigma}_{b}^{2} = \sigma _{b}^{2} = 0\). This implies that \(V^{\star }_{0}(\Lambda )^{3}=0\) since \(V^{\star }_{0}(\Lambda )\) has no constant term and has at least one least source field in each of its two summands. Given \(V^{\star }_{0}\), we inductively define renormalised interaction potentials that share this property:
where
and \(u_{j+1}^{\star }{(\Lambda )}\) collects the terms that do not contain \(\psi \) or \(\bar{\psi}\). Consequently, one can check that
For convenience, in the last step when \(j=N\), we set \(C_{N+1}=t_{N}Q_{N}\). This separation of the zero mode is not essential here but will be useful for our analysis in the interacting case.
For \(j>0\), the \(V^{\star }_{j}\) have terms not present in \(V^{\star }_{0}\), for example the terms involving \(q\) in the next definition. The nilpotency of the source fields \(\sigma _{a}\), \(\sigma _{b}\) limits the possibilities.
Definition 5.1
Let \(\mathcal {V} ^{\star }\) be the space of formal field polynomials \(u^{\star }+V^{\star }\) of the form:
for observable coupling constants \((\lambda _{a},\lambda _{b},q, r)\in \mathbb{C}^{4}\) respectively \((\lambda _{a},\lambda _{b},\gamma _{a},\gamma _{b},q, \eta , r)\in \mathbb{C}^{7}\). For \(X \subset \Lambda \), we define \((u^{\star }+V^{\star })(X) \in \mathcal {N} ^{\star }(X \cap \{a,b\})\) by
in Case (1), and analogously in Case (2).
Remark 5.2
The terms corresponding to \(r\) do not appear at any step of the free observable flow (5.6) if we start with them equal to 0. We include them here in preparation for the interacting model.
The evolutions of \(u^{\star }_{j}+V^{\star }_{j} \to u^{\star }_{j+1}\) and \(V^{\star }_{j} \to V^{\star }_{j+1}\) are equivalent to the evolution of the coupling constants \((\lambda _{a},\lambda _{b},q, r)\) respectively \((\lambda _{a},\lambda _{b},\gamma _{a},\gamma _{b},q,\eta ,r)\). By computation of the fermionic Gaussian moments in (5.6), the flow of the observable coupling constants according to (5.6) is then given as follows. Note that the evolution of coupling constants in \(V^{\star }\) is independent of the coupling constants in \(u^{\star }\).
Lemma 5.3
Let \(V^{\varnothing }_{0}=0\), and let \(u^{\star }_{j}\) and \(V^{\star }_{j}\) be of the form as in Definition 5.1. The map (5.6) is then given as follows. In Case (1), for \(x\in \{a,b\}\),
whereas in Case (2), for \(x\in \{a,b\}\),
Proof
This follows from straightforward evaluation of (5.6) using (3.10). □
To continue the warm-up for the interacting case, we illustrate how these equations reproduce the direct computations of correlation functions (and explain the terminology of observable coupling constants). When \(r_{0}=0\), by a computation using Definition 5.1, the formulas (5.2) and (5.4)–(5.5) imply the correlation functions in Cases (1) and (2) are given by
with \(q=q^{(1)}\) and \(\lambda =\lambda ^{(1)}\) for the first equation and \(q=q^{(2)}\), \(\gamma =\gamma ^{(2)}\), and \(\lambda =\lambda ^{(2)}\) for the last two. Recalling the convention \(C_{N+1}=t_{N}Q_{N}\), for Case (2) with \(r_{0}=0\) a computation using (5.13), (5.15) and (5.16) shows
with the final equality by (3.1). Combined with (5.18), this gives
as expected for free fermions.
Remark 5.4
In the preceding computation we kept the potential in the exponential for the entire computation, whereas in Sects. 4 and 6 the zero mode is integrated out directly without rewriting the integrand in this form (see, e.g., (4.3)). We distinguish these two approaches by using \(N+1\) subscripts for the former and \((N,N)\) for the latter, and by putting tildes on quantities associated with the \((N,N)\)th step as was done in Sect. 4.
Before moving to the interacting model, we introduce the coalescence scale \(j_{ab}\) as the largest integer \(j\) such that \(C_{k+1}(a,b)=0\) for all \(k < j\), i.e.,
In the degenerate cases \(\lambda _{a}=0\) or \(\lambda _{b}=0\) when only one of the source fields is present we use the convention \(j_{ab}=+\infty \). Note that the finite range property (3.2) implies that \(q_{j}=\eta _{j}=r_{j}=0\) for \(j< j_{ab}\) provided they are all 0 when \(j=0\). This will also be true in the interacting case.
In connection with the coalescence scale, we also make a convenient choice of the block decomposition of \(\Lambda _{N}\) based on the relative positions of \(a\) and \(b\). Namely, we center the block decomposition such that point \(a\) is in the center (up to rounding if \(L\) is even) of the blocks at all scales \(1 \leqslant j \leqslant N\). This implies that if \(|a-b|_{\infty }< \frac{1}{2} L^{j+1}\) the scale-\(j\) blocks containing \(a\) and \(b\) are contained in a common scale-\((j+1)\) block.
5.3 Norms with observables
To extend the above computation for \(V^{\varnothing }=0\) to the interacting case, we will extend the renormalisation group map to the Grassmann algebra amended by the source fields. In Case (2), recall that the source fields \(\sigma _{a}\) and \(\sigma _{b}\) are even elements rather than Grassmann generators themselves, i.e., they are commuting elements also satisfying \(\sigma _{a}^{2}=\sigma _{b}^{2}=0\). In both Cases (1) and (2), this algebra has the decomposition
where \(\mathcal {N} ^{\varnothing}(X)\) is spanned by monomials with no factors of \(\sigma \), \(\mathcal {N} ^{a}(X)\) is spanned by monomials containing a factor \(\sigma _{a}\) but no factor \(\bar{\sigma}_{b}\) (respectively \(\sigma _{b}\)), analogously for \(\mathcal {N} ^{b}(X)\), and \(\mathcal {N} ^{ab}(X)\) is spanned by monomials containing \(\sigma _{a}\bar{\sigma}_{b}\) respectively \(\sigma _{a}\sigma _{b}\). Thus any \(F \in \mathcal {N} (X)\) can be written as
with \(F_{\varnothing},F_{a},F_{b},F_{ab} \in \mathcal {N} ^{ \varnothing}(X)\). We denote by \(\pi _{\varnothing}\), \(\pi _{a}\), \(\pi _{b}\) and \(\pi _{ab}\) the projections on the respective components, e.g., \(\pi _{a}F =\sigma _{a}F_{a}\), and \(\pi _{\star }= \pi _{a}+\pi _{b}+\pi _{ab}\). We will use superscripts instead of subscripts in the decomposition when the factors of \(\sigma \) are included, e.g., \(F^{a}= \sigma _{a}F_{a}\) and \(F^{\varnothing}=F_{\varnothing}\).
We say that \(F\) is \(U(1)\) invariant if the number of generators with a bar is equal to the number without a bar. Explicitly, in Case (1) this means \(F_{\varnothing} \) and \(F_{ab}\) are \(U(1)\) invariant, \(F_{a}\) has one more factor with a bar than without, and similarly for \(F_{b}\). In Case (2) this means all of \(F_{\varnothing}\), \(F_{a}\), \(F_{b}\) and \(F_{ab}\) are \(U(1)\) invariant (recall that \(\sigma _{a}=\bar{\vartheta}_{a}\vartheta _{a}\) and \(\sigma _{b}=\bar{\vartheta}_{b}\vartheta _{b}\)). Denote by \(\mathcal {N} _{\mathrm{sym}}(X)\) the subalgebra of \(U(1)\) invariant elements.
For \(F\) decomposed according to (5.23) we define
where
In particular, \(\|\sigma _{a}\|_{T_{j}(\ell _{j})} = \ell _{a,j}\) and \(\|\sigma _{a}\sigma _{b}\|_{T_{j}(\ell _{j})} = \ell _{ab,j}\) and, in Cases (1) and (2), respectively,
and, again in the two cases respectively,
In both cases these terms do not change size under change of scale, provided that \(j\geqslant j_{ab}\) for the last term. Thus they are marginal. As will be seen in Sect. 6, see the paragraph following Lemma 6.3, the choices of \(\ell _{a,j}\) and \(\ell _{ab,j}\) are appropriate to capture the leading behaviour of correlation functions.
The extended definition (5.24) of the \(T_{j}(\ell _{j})\) norm satisfies the properties discussed in Sect. 3.5, with the exception of the generalisation of the monotonicity estimate \(\|F_{\varnothing}\|_{T_{j+1}(2\ell _{j+1})} \leqslant \|F_{ \varnothing}\|_{T_{j}(\ell _{j})}\). Checking these properties is straightforward by using the properties of the bulk norm, and, in the case of the product property, using that \(\ell _{ab,j}\leqslant \ell _{a,j}\ell _{b,j}\) (recall (3.23)). Similar reasoning also yields a weaker monotonicity-type estimate: by (5.24), (3.23), and monotonicity in the bulk algebra,
5.4 Localisation with observables
We combine the space \(\mathcal {V} ^{\varnothing }\) of bulk coupling constants from Definition 3.5 with the space \(\mathcal {V} ^{\star }\) of observable coupling constants from Definition 5.1 into
We extend the localisation operators \(\operatorname{Loc}_{X,Y}\) from Sect. 3.6 to the amended Grassmann algebra (5.22) as follows. As in the bulk setting, we will focus on the key properties of the extended localisation operators. The extension of \(\operatorname{Loc}_{X,Y}\) is linear and block diagonal with respect to the decomposition (5.22), and so can be defined separately on each summand. On \(\mathcal {N} ^{\varnothing}(X)\), the restriction \(\operatorname{Loc}_{X,Y}\) is defined to coincide with the operators from Proposition 3.8. From now on we denote this restriction by \(\operatorname{Loc}_{X}^{\varnothing}\) or \(\operatorname{loc}^{\varnothing}_{X}\) if we want to distinguish it from the extended version. To define the restriction \(\operatorname{Loc}^{\star }_{X,Y}\) of \(\operatorname{Loc}_{X,Y}\) to \(\mathcal {N} ^{\star }(X)\), we continue to employ the systematic framework from [30, Sect. 1.7], as follows. Let \(\operatorname{loc}^{a}\), \(\operatorname{loc}^{b}\), and \(\operatorname{loc}^{ab}\) be the localisation operators from [30, Definition 1.17] with maximal dimensions (for the Cases \((p=1)\) and \((p=2)\))
Case (1). For \(\sigma _{a} F_{a}\in \mathcal {N} ^{a}(X)\) we set \(\operatorname{Loc}_{X, Y} (\sigma _{a} F_{a}) = \sigma _{a} \operatorname{loc}^{a}_{X\cap \{a\}, Y\cap \{a\}} F_{a}\), and likewise for point \(b\). For \(\sigma _{a}\bar{\sigma}_{b} F_{ab} \in \mathcal {N} ^{ab}(X)\) we set \(\operatorname{Loc}_{X, Y} (\sigma _{a}\bar{\sigma}_{b} F_{ab}) = \sigma _{a} \bar{\sigma}_{b} \operatorname{loc}^{ab}_{X\cap \{a,b\}, Y \cap \{a,b\}} F_{ab}\).
Case (2). The definitions in Case (2) are analogous, but with \(\bar{\sigma}_{b}\) replaced by \(\sigma _{b}\).
The superscripts \(\varnothing \), \(a\), \(b\), \(ab\) are present to indicate that we have assigned different maximal dimensions to the summands in (5.22). We use the same choice of field dimensions \([\psi ]=[\bar{\psi}]=(d-2)/2\) as in Sect. 3.6. We note that \(\operatorname{loc}^{a}_{X, \varnothing}=\operatorname{loc}^{b}_{X, \varnothing}=\operatorname{loc}^{ab}_{X,\varnothing}=0\). The main difference between these operators and \(\operatorname{Loc}^{\varnothing}\) is that the expressions produced by \(\operatorname{loc}^{a}\), \(\operatorname{loc}^{b}\), \(\operatorname{loc}^{ab}\) are local, i.e., supported near \(a\) and \(b\). A second difference is that the maximal dimensions vary.
Before giving the general properties of the extended \(\operatorname{Loc}\) we include an example in Case (1).
Example 5.5
Consider Case (1). Then:
(i) If \(F\in \mathcal {N} (\Lambda )\) is a field monomial of degree greater than one or if \(F\) has gradients in it, then \(\operatorname{Loc}^{a}_{X} F = \operatorname{Loc}^{b}_{X} F=0\). Here (and in the rest of this example) we do not count factors of \(\sigma _{a}\) and \(\sigma _{b}\) in the degree. If \(F\) has degree greater than 2 or is degree two and has gradients \(\operatorname{Loc}^{ab}_{X} F =0\).
(ii) If \(F= \sigma _{a} \bar{\psi}_{x}+ \bar{\sigma}_{b} \psi _{y}+ \sigma _{a} \bar{\sigma}_{b} \bar{\psi}_{x}\psi _{y}\), then
The next proposition summarises the key properties of the operators \(\operatorname{Loc}_{X,Y}\). As with Proposition 3.8, these properties follow from [30]. That the choice of maximal dimensions (5.30) produce contractive estimates can intuitively be understood by considering the marginal monomials. By (5.26) and (5.27), these are exactly the monomials with dimensions \(d_{+}^{a}=d_{+}^{b}\) respectively \(d_{+}^{ab}\).
Proposition 5.6
For \(L=L(d)\) sufficiently large there is a universal \(\bar{C}>0\) such that: for \(j< N\) and any small sets \(Y \subset X \in \mathcal {S} _{j}\), the linear maps \(\operatorname{Loc}_{X,Y}^{\star }\colon \mathcal {N} ^{ \star }(X^{\square}) \to \mathcal {N} ^{\star }(Y^{\square})\) have the following properties:
(i) They are bounded:
(ii) For \(j\geqslant j_{ab}\), the maps \(\operatorname{Loc}_{X}^{\star } =\operatorname{Loc}_{X,X}^{\star } \colon \mathcal {N} ^{\star }(X^{\square}) \to \mathcal {N} ^{\star }(X^{\square})\) satisfy the contraction bound:
Moreover, the bound (5.33) holds also for \(j< j_{ab}\) if \(F^{ab}=0\).
(iii) If \(X\) is the disjoint union of \(X_{1}, \dots , X_{n}\) then \(\operatorname{Loc}_{X}^{\star } = \sum _{i=1}^{n} \operatorname{Loc}_{X,X_{i}}^{ \star }\).
(iv) For a block \(B\) and polymers \(X \supset B\), \(\operatorname{Loc}_{X,B}^{\star }F \in \mathcal {V} ^{ \star }(B)\) if \(F \in \mathcal {N} ^{\star }_{\mathrm{sym}}(X)\).
Properties (i)–(iii) follow from [30] in the same way as the corresponding properties in Proposition 3.8 by making use of the observation that
in Case (2) and analogously in Case (1). These factors of \(L^{d_{+}}\) correspond to the missing \(L^{-d_{+}}\) factors in (5.33) as compared to Proposition 3.8. It only remains to verify (iv), i.e., to identify the image of \(\operatorname{Loc}_{X,B}^{\star }\) when acting on \(F \in \mathcal {N} ^{\star }_{\mathrm{sym}}(X)\).
Case (1). By the choice of dimensions in its specification, the image of \(\sigma _{a}\operatorname{loc}^{a}\) is spanned by the local monomials \(\sigma _{a}\), \(\sigma _{a}\bar{\psi}_{a}\), \(\sigma _{a}\psi _{a}\). The condition of \(U(1)\) invariance then implies that if \(\sigma _{a} F_{a}\in \mathcal {N} ^{a}_{\mathrm{sym}}(X)\) only the monomial \(\sigma _{a}\bar{\psi}_{a}\) is admissible. The situation is analogous for \(\operatorname{loc}^{b}\). Similarly, \(\sigma _{a}\bar{\sigma}_{b}\operatorname{loc}^{ab}\) has image spanned by \(\sigma _{a}\bar{\sigma}_{b}\) and \(\sigma _{a}\bar{\sigma}_{b}\bar{\psi}_{x}\psi _{x}\) for \(x\in \{a,b\}\) as well as further first order monomials with at most \((d-2)/2\) gradients, e.g., \(\sigma _{a}\bar{\sigma}_{b}\nabla _{e_{1}}\psi _{x}\). Only the even monomials \(\sigma _{a}\bar{\sigma}_{b}\), \(\sigma _{a}\bar{\sigma}_{b}\bar{\psi}_{a}\psi _{a}\), and \(\sigma _{a}\bar{\sigma}_{b}\bar{\psi}_{b}\psi _{b}\) are compatible with \(U(1)\) symmetry. In summary, \(\operatorname{Loc}^{\star }_{X,Y} F\) is contained in \(\mathcal {V} ^{\star }\) if \(F\in \mathcal {N} ^{\star }_{\mathrm{sym}}(X)\).
Case (2). By the choice of dimensions, in this case \(\sigma _{a}\operatorname{loc}^{a}\) has image spanned by the local monomials \(\sigma _{a}\), \(\sigma _{a}\bar{\psi}_{a}\psi _{a}\) as well as further first order monomials with at most \(d-2\) gradients, and \(U(1)\) symmetry implies that only the even terms \(\sigma _{a}\) and \(\sigma _{a}\bar{\psi}_{a}\psi _{a}\) arise in the image if \(F \in \mathcal {N} _{\mathrm{sym}}(X)\). The analysis for \(\sigma _{b}\operatorname{loc}^{b}\) is analogous. Lastly, \(\sigma _{a}\sigma _{b}\operatorname{loc}^{ab}\) has image spanned by \(\sigma _{a}\sigma _{b}\) and the monomials \(\sigma _{a}\sigma _{b}\bar{\psi}_{x}\psi _{x}\) for \(x\in \{a,b\}\) and first order monomials with at most \((d-2)/2\) gradients. Again only the even monomials are compatible with \(U(1)\) symmetry.
5.5 Definition of the renormalisation group map with observables
In this section the renormalisation group map \(\Phi _{j+1} =\Phi _{j+1,N,m^{2}}\) is extended to include the observable components (as in Sect. 3, we omit the \(N\) and \(m^{2}\)-dependence when there is no risk of confusion). To this end, we now call the renormalisation group map from Sect. 3.8 the bulk component and denote it by \(\Phi ^{\varnothing}_{j+1}\), and \(\Phi _{j+1}=(\Phi ^{\varnothing}_{j+1},\Phi ^{\star }_{j+1})\) will now refer to the renormalisation group map extended to the algebra with observables. The map \(\Phi ^{\star }_{j+1}\) is the observable component of the renormalisation group map. This extension will be defined so that the bulk components of \(K_{j+1}\) and \(V_{j+1}\) only depend on the bulk components of \(K_{j}\) and \(V_{j}\). In other words,
On the other hand, the observable components \(V^{\star }_{j+1}\) and \(K^{\star }_{j+1}\) will depend on both the observable and the bulk components of \((V_{j},K_{j})\). The observable component \(\Phi ^{\star }_{j+1}\) is upper-triangular in the sense that the \(a\) component \(\pi _{a}\Phi ^{\star }_{j+1}(V_{j},K_{j})\) of \(\Phi ^{\star }_{j+1}(V_{j},K_{j})\) only depends on \((V^{\varnothing }_{j},K^{\varnothing }_{j})\) and \((V^{a}_{j},K^{a}_{j})\) but not on \((V^{b}_{j},K^{b}_{j})\) or \((V^{ab}_{j},K^{ab}_{j})\), and similarly for the \(b\) component. The \(ab\) component depends on all components from the previous scale. We will use an initial condition \(V_{0}\in \mathcal {V} \) and \(K_{0}(X)=1_{X=\varnothing}\).
We now give the precise definition of the observable component of the renormalisation group map \(\Phi _{j+1}^{\star }\colon (V_{j},K_{j})\mapsto (u^{\star }_{j+1}, V^{ \star }_{j+1}, K^{\star }_{j+1})\). For \(j+1< N\), given \((V_{j}, K_{j})\) and \(B \in \mathcal {B} _{j}\), define \(Q(B)\) and \(J(B,X)\) as in (3.44)–(3.46) using the extended version of \(\operatorname{Loc}\) from Sect. 5.4. If \(j+1=N\) set \(Q=J=0\). We let \(Q^{\star }(B)=\pi _{\star }Q(B) \) and \(J^{\star }(B,X)=\pi _{\star }J(B, X)\) denote the observable components. The new detail for the observable renormalisation group map is that, to define \(V^{\star }_{j+1}\), we include the second order contribution from \(V^{\star }_{j}\) in order to maintain better control on the renormalisation group flow. To this end, for \(j+1\leqslant N\) and \(B,B'\in \mathcal {B} _{j}\), let
The following observations will be useful later. Since \(V^{\star }(B), Q^{\star }(B) \in \mathcal {V} ^{\star }(B)\), the sum over \(B'\) contains at most two non-zero terms, corresponding to the blocks containing \(a\) and \(b\). Since the covariance matrix \(C_{j+1}\) has the finite range property (3.2), also \(P^{\star }(B,B')=0\) for \(B\neq B'\) if \(|a-b|_{\infty }\geqslant \frac{1}{2} L^{j+1}\). Finally, if \(a\) and \(b\) are not in the same block, then \(P^{\star }(B,B)=0\) since the source fields square to zero.
With these definitions in place, \(u^{\star }_{j+1}+V^{\star }_{j+1}\) is defined in the same way as \(u_{j+1}+ V_{j+1}\) with the addition of the second order term \(P^{\star }\), and \(K^{\star }_{j+1}\) is then defined in the same way as \(K_{j+1}\):
Definition 5.7
The map \((V_{j},K_{j}) \mapsto ({ u^{\star }_{j+1}},V^{\star }_{j+1})\) is defined, for \(B\in \mathcal {B} _{j}\), by
where \(u^{\star }_{j+1}\) consists of all monomials that do not contain factors of \(\psi \) or \(\bar{\psi}\). Explicitly,
The map \((V_{j},K_{j})\mapsto K^{\varnothing }_{j+1}+K^{\star }_{j+1}\) is defined by the same formula as in Definition 3.10 except that \(V^{\varnothing}\) and \(u^{\varnothing}\) are replaced by \(V=V^{\varnothing}+V^{\star }\) and \(u=u^{\varnothing}+u^{\star }\).
Propositions 3.11 and 3.12 also hold for this extended definition of the renormalisation group map. The proofs are the same as presented in Sect. 3.7.
5.6 Estimates for the renormalisation group map with observables
In this section, the \(O\)-notation refers to scale \(j+1\) norms, i.e., for \(F,G \in \mathcal {N} (\Lambda )\), we write \(F=G+O(t)\) to denote that \(\|F-G\|_{T_{j+1}(\ell _{j+1})} \leqslant O(t)\). We use \(\|V_{j}\|_{j}\) and \(\|K_{j}\|_{j}\) defined in (3.64)–(3.65), with the understanding that the right-hand sides of the definitions are the norm (5.24) which accounts for source fields.
Theorem 5.8
Under the assumptions of Theorem 3.13, if also \(\|V^{\star }_{j}\|_{j}+\|K^{\star }_{j}\|_{j} \leqslant \varepsilon \) and \(u^{\star }_{j}=0\), then for \(j+1< N\) the observable components of the renormalisation group map \(\Phi ^{\star }_{j+1}\) satisfy
provided that \(K^{ab}_{j}(X)=0\) for \(X\in \mathcal {S} _{j}\) if \(j< j_{ab}\). Both \(\eta =\eta (d)\) and \(\nu =\nu (d)\) are positive geometric constants. For \(j+1=N\), \(\Phi ^{\star }_{N}\) is bounded.
The first estimate in the theorem expresses that the evolution of \(u^{\star }+V^{\star }\) is given by second-order perturbation theory, plus a higher order remainder due to \(K^{\star }\). The second estimate states that \(K^{\star }\) is contracting (for \(L\) and \(A\) large), up to error terms at most as large as the bulk coupling constants \(V^{\varnothing }\) and \(K=K^{\varnothing }+K^{\star }\). The additional factor \(\|V_{j}\|_{j}+\|K_{j}\|_{j} \geqslant \|V^{\star }_{j}\|_{j}\) will be small (but of order 1) while all other coordinates will be exponentially small in \(j\). Indeed, as a consequence of the above theorem, Proposition 5.15 below states that if the bulk flow \((V^{\varnothing },K^{\varnothing })\) is as constructed in Sect. 3.9 then \(V^{\star }\) remains bounded while \(K^{\star }\) goes to 0 exponentially fast.
The proof of the theorem follows that of Theorem 3.13 closely, with improvements for the leading terms that allow for \(V^{\star }\) to be tracked to second order. It is given in the remainder of this subsection. The reader may again wish to skip the details of this proof on a first reading and proceed to the application of these estimates Sect. 5.7.
5.6.1 Coupling constants
We first give a bound on \(u^{\star }_{j+1}(\Lambda )+ V^{\star }_{j+1}(\Lambda )\). By Proposition 5.6 (iii),
Since only small sets \(X\) that contain \(a\) or \(b\) contribute, Proposition 5.6 (i) implies
By algebraic manipulation, the product property, that \(\mathbb{E}_{C_{j+1}}\theta \) is a contraction, (5.28), and (5.43),
Putting these pieces together establishes (5.40) as \(L^{2(d-2)}(\|V_{j}\|_{j}+\|K_{j}\|_{j})\leqslant 1\) if \(\varepsilon =\varepsilon (L)\) is small enough. An immediate consequence is
The same bounds hold with \(\Lambda \) replaced by any \(X\in \mathcal {P} _{j}\). These will be used in the following analysis.
5.6.2 Small sets
The most significant improvement in the analysis concerns small sets, which we now analyse to second order. To simplify notation, we write
Lemma 5.9
For any \(B,B' \in \mathcal {B} _{j}\),
Proof
Note that \(P^{\star }(B,B')=\frac{1}{2}\mathbb{E}_{C_{j+1}}[{\theta \hat{V}^{ \star }_{j}(B);\theta \hat{V}^{\star }_{j}(B')}]\). Since it is quadratic in \(\hat{V}^{\star }_{j}\in \mathcal {V} ^{\star }\), \(P^{\star }(B,B')\) can only contain monomials with a factor of \(\sigma _{a}\bar{\sigma}_{b}\) (Case (1)) or \(\sigma _{a}{\sigma _{b}}\) (Case (2)) because \(\sigma _{a}^{2}=\sigma _{b}^{2}=\bar{\sigma}_{b}^{2}=0\). Similarly, for any \(W \in \mathcal {V} ^{\star }\) and \(B,B',B'' \in \mathcal {B} _{j}\), it follows that \(P^{\star }(B,B') W (B'')=0\). The claim follows as this implies that \((\mathbb{E}_{C_{j+1}}[\theta \hat{V}^{\star }_{j}(B)])(\mathbb{E}_{C_{j+1}}[ \theta \hat{V}^{\star }_{j}(B')])\) is the same as
□
The next lemmas are analogues of Lemmas 3.16–3.17 that apply to the observable components. We begin with the replacement for Lemma 3.17. For \(B\in \mathcal {B} _{j}\), recall \(\bar{B}\) denotes the scale \(j+1\)-block containing \(B\).
Lemma 5.10
Suppose that \(\|V_{j}\|_{j}+\|K_{j}\|_{j}\leqslant 1\). Then for any \(X \in \mathcal {P} _{j}\), denoting by \(n\in \{0,1,2\}\) the number of \(B\in \mathcal {B} _{j}(X)\) containing \(a\) or \(b\),
For any \(B\in \mathcal {B} _{j}\) such that \(\bar{B}\) contains at most one of \(a\) and \(b\),
Moreover if \(|a-b|_{\infty }\geqslant \frac{1}{2} L^{j+1}\) then for any \(X\in \mathcal {P} _{j}\) with \(|\mathcal {B} _{j}(X)|=2\),
Proof
Throughout the proof, we will use that for \(V\) representing either \(V_{j}\) or \(u_{j+1}+V_{j+1}\) one has
where we recall that the \(O\)-notation refers to terms whose \(T_{j+1}(\ell _{j+1})\)-norms are bounded by the indicated numbers, up to multiplicative constants. For both of the choices for \(V\), one has \(\|V^{\varnothing }(B) \|_{T_{j+1}(\ell _{j+1})}\leqslant \|V_{j}^{ \varnothing}\|_{j}+O(\|K_{j}^{\varnothing}\|_{j}) \leqslant O(1)\) by (3.71) and \(\|V^{\star }(B)\|_{T_{j+1}(\ell _{j+1})}\leqslant O(\|V_{j}^{\star } \|_{j}+L^{2(d-2)}\|K_{j}^{\star }\|_{j})\) by using (5.45)–(5.46) (with \(B\) instead of \(\Lambda \)).
To show (5.50), for each \(B\in \mathcal {B} _{j}\), write \(\delta I(B) = \pi _{\varnothing}\delta I(B) + \pi _{\star }\delta I(B)\) and expand the product defining \((\delta I)^{X}\) using that there are \(n\) blocks \(B\) for which \(\pi _{\star }\delta I(B)\neq 0\). The claim then follows since \(\|\pi _{\varnothing} \delta I(B)\|_{T_{j+1}(\ell _{j+1})} \leqslant O( \|V^{\varnothing }_{j}\|_{j} + \|K^{\varnothing }_{j}\|_{j})\) by Lemma 3.17 and \(\|\pi _{\star }\delta I(B)\|_{T_{j+1}(\ell _{j+1})}=O(\|V^{\star }_{j} \|_{j}+L^{2(d-2)}\|K^{\star }_{j}\|_{j})\) which follows from the previous paragraph (as the doubling map commutes with \(\pi _{\star }\)).
For the bound (5.51), using that \(B\) can contain only \(a\) or \(b\) by assumption and that source fields square to zero, one has \(V^{\star }(B)^{2}=0\) for \(V\) either \(V_{j}\) or \(u_{j+1}+V_{j+1}\). Thus (5.53) simplifies to
Observe that \(P^{\star }(B)=0\) since \(\bar{B}\) contains only one of \(a\) and \(b\), see the remark below (5.37). As a result, (5.38) and the above show that the term linear in \(V_{j}^{\star }(B)\) in \(\pi _{\star }\mathbb{E}_{C_{j+1}}\delta I(B)\) cancels in expectation. The claim (5.51) then follows from \(\|\mathbb{E}_{C_{j+1}}\theta Q^{\star }(B)\|_{T_{j+1}(\ell _{j+1})}=O(L^{2(d-2)} \|K_{j}^{\star }\|_{j})\) by (5.43) and (5.28), and bounding the quadratic terms using (3.71) and (5.45)–(5.46) as below (5.53).
For the final assertion (5.52), we first show that \(\mathbb{E}_{C_{j+1}}(\pi _{\star }\delta I)^{X}=L^{4(d-2)} O(\|V^{ \varnothing }_{j}\|_{j} + \|K_{j}\|_{j})(\|V^{\star }_{j}\|_{j} + \|K^{ \star }_{j}\|_{j})\), where we emphasise that \(\pi _{\star }\) is inside the product over \(X\). To see this bound, let \(X = B\cup B'\), and note that \(V^{\star }(B)\) and \(V^{\star }(B')\) are either 0 or polynomials in \(\psi _{a}\), \(\bar{\psi}_{a}\) and \(\psi _{b}\), \(\bar{\psi}_{b}\) respectively. Since by assumption \(C_{j+1}(a, b)=0\), \(\mathbb{E}_{C_{j+1}}\theta V^{\star }(B) V^{\star }(B') =0\). Hence a nonvanishing contribution to \(\mathbb{E}_{C_{j+1}}(\pi _{\star }\delta I)^{X}\) involves at least one factor \(V^{\varnothing }\) from the expansion of the \(\delta I\) by (5.53). The factor of \(L^{4(d-2)}\) arises from applying (5.28). The estimate (5.52) now follows similarly to the previous cases:
as the cross terms with one factor \(\pi _{\star }\) and one factor \(\pi _{\varnothing}\) satisfy this bound as above. □
Next we replace Lemma 3.16. Unlike before we explicitly consider terms arising from two blocks, in order to obtain a cancellation up to a third order error in \(V^{\star }\). Indeed, note that the right-hand side of (5.56) involves \(\|V_{j}\|_{j}\|V^{\varnothing }_{j}\|_{j} \leqslant (\|V^{ \varnothing }_{j}\|_{j}+\|V^{\star }_{j}\|_{j})\|V^{\varnothing }_{j} \|_{j}\) but no term \(\|V^{\star }_{j}\|_{j}^{2}\), and that \(\|V^{\varnothing }_{j}\|_{j}\) is exponentially small in \(j\) along the flow while \(\|V^{\star }_{j}\|_{j}\) is of order 1. The \(K\)-terms are higher order.
Lemma 5.11
Suppose that \(\|V_{j}^{\varnothing }\|_{j}+\|K_{j}^{\varnothing}\|_{j}\leqslant \varepsilon \) and \(\|V^{\star }_{j}\|_{j}+\|K^{\star }_{j}\|_{j}\leqslant \varepsilon \). Then for \(B\in \mathcal {B} _{j}\),
Proof
Recall \(\tilde{V}^{\star }_{j+1}=u^{\star }_{j+1}+V^{\star }_{j+1}\). Using (5.38) to re-express \(\mathbb{E}_{C_{j+1}}[\theta Q^{\star }(B)]\), the term inside the norm on the left-hand side of (5.56) equals
We start with the one block terms \(B'=B\) in (5.57). Using Lemma 5.9 to rewrite \(P(B,B)\) and since \(\delta I(B)=\theta e^{-V_{j}(B)}-e^{-(V_{j+1}+u_{j+1})(B)}\), these terms are
To estimate these terms, first note that if \(V=V_{j+1}+u_{j+1}\) then (5.40) and its consequences (5.45)–(5.46) imply \(\|V^{\varnothing }\|_{T_{j+1}(\ell _{j+1})}\leqslant 1\), \(\|V^{\star } \|_{T_{j+1}(\ell _{j+1})}\leqslant 1\). This bound also holds for \(V=V_{j}\) provided \(\varepsilon \) is sufficiently small by (5.28), we then have for \(V=V_{j}\) or \(V=u_{j+1}+V_{j+1}\),
where we have used \(V^{\star }(B)^{3}=0\), and in the case \(V=u_{j+1}+V_{j+1}\), (3.71) to control \(\|u^{\varnothing}_{j+1}+V^{\varnothing }_{j+1}\|_{j+1}\) in terms of \(\|V^{\varnothing }_{j}\|_{j}+\|K^{\varnothing }_{j}\|_{j}\) and (5.46) to control \(\|u^{\star }_{j+1}+V^{\star }_{j+1}\|_{j+1}\) similarly. Using also
by the product property, (5.43), (5.28), and the assumed norm bounds, the estimate for the one block terms follow.
Recall that \(P^{\star }(B,B')=0\) unless \(a\), \(b\) are each in one of the two blocks. Thus for \(B' \neq B\) the two block terms in (5.57) are, by Lemma 5.9,
We start by rewriting this in a more convenient form. Let \(\delta V_{j}^{\star } = \theta \hat{V}_{j}^{\star }-\tilde{V}_{j+1}^{ \star }\). By (5.38), \(\mathbb{E}_{C_{j+1}}\theta \hat{V}^{\star }_{j} = \tilde{V}^{\star }_{j+1} + P^{\star }= \tilde{V}^{\star }_{j+1} + O(\sigma _{a}\sigma _{b})\), where \(O(\sigma _{a}\sigma _{b})\) denotes a monomial containing a factor \(\sigma _{a}\bar{\sigma}_{b}\) in Case (1) or a factor \(\sigma _{a}\sigma _{b}\) in Case (2). Since all terms in \(\delta V^{\star }_{j}\) contain a source field (that is, a \(\sigma \)-factor) and source fields square to zero, we obtain
Therefore we need to estimate
First write
The second and third terms on the right-hand side are \(O((\|V^{\star }_{j}\|_{j}+L^{2(d-2)}\|K^{\star }_{j}\|_{j}) (\|V^{ \varnothing }_{j}\|_{j}+\|K^{\varnothing }_{j}\|_{j}))\) using Lemma 3.17 for \(\pi _{\varnothing }\delta I\) and \(\|\pi _{\star }\delta I(B)\|_{T_{j+1}(\ell _{j+1})} =O(\|V^{\star }_{j} \|_{j}+L^{2(d-2)}\|K^{\star }_{j}\|_{j})\) by (5.46). Using (5.59), the term \(\pi _{\star }\delta I(B)\pi _{\star }\delta I(B')\) can be estimated as
since \(\sigma _{a}^{2}=\sigma _{b}^{2}=\bar{\sigma}_{b}^{2}=0\). The factor \(L^{4d}\) is a convenient common bound. □
The next lemma replaces Lemma 3.15 on the observable components.
Lemma 5.12
For any \(U \in \mathcal {C} _{j+1}\), if \(K^{ab}_{j}(Y)=0\) for all \(Y\in \mathcal {S} _{j}\) and all \(j< j_{ab}\), then
Proof
The proof is the same as that of Lemma 3.15 except for the following observation. The sum over \(X \in \mathcal {S} _{j}\) that contributes a factor \(O(L^{d})\) in the proof of Lemma 3.15 only contributes \(O(1)\) on the observable components because for these only the small sets containing \(a\) or \(b\) contribute. Thus the bound for \(\operatorname{Loc}^{\star }\) from Proposition 5.6, which lacks a factor \(L^{-d}\) compared to the bound for \(\operatorname{Loc}^{\varnothing}\), produces the same final bound. □
Proof of Theorem 5.8
The proof is analogous to that of Theorem 3.13, and we proceed in a similar manner, by beginning with the coupling constants and then an estimate of \(\pi _{\star }\mathcal {L} _{j+1}(U)\), where \(\mathcal {L} _{j+1}(U)\) is defined by the formula (3.72) but with the extended coordinates introduced in Sect. 5.5.
For the coupling constants, i.e., the analogue of Sect. 3.8.1, the bound (3.69) gets replaced by (5.43) which gives \(\|Q^{\star }(B)\|_{T_{j+1}(\ell _{j+1})}\leqslant O(L^{d-2}\|K^{ \star }_{j}\|)\), and we also have \(\|u_{j+1}^{\star }\|_{j+1}+\|V^{\star }_{j+1}\|_{j+1}\leqslant O(\|V^{ \star }_{j}\|+L^{d-2}\|K^{\star }_{j}\|_{j})\) by (5.45)–(5.46).
Note that the terms of \(\mathcal {L} _{j+1}(U)\) are of the form \(\sum _{F}e^{-V_{j+1}(U\setminus X)+u_{j}|X|}F\). We will use that
We first explain how to estimate the sum arising from the first term, which only contributes to the second term in the estimate. The estimation of the terms \(\pi _{\varnothing}F\) is exactly as in Sect. 3.8. Estimating \(\pi _{\star }e^{-V_{j+1}(U\setminus X)+u_{j}|X|}\), and the resultant sum over \(F\), requires a replacement of Lemma 3.18. For this it suffices to note that \(\| \pi _{\star }e^{-V_{j+1}(U\setminus X)+u_{j+1}|X|}\|_{T_{j+1}( \ell _{j+1})}\leqslant (\|V^{\star }_{j}\|_{j}+L^{2(d-2)}\|K^{\star }_{j} \|_{j}) 2^{|\mathcal {B} _{j}(X)|}\). This estimate follows by the product property, (5.45)–(5.46), and arguing as in the proof of Lemma 3.18. Hence this term is bounded by \(O(\|V^{\star }_{j}\|_{j}+L^{2(d-2)}\|K^{\star }_{j}\|_{j})(\|K^{ \varnothing }_{j}\|_{j} + A^{\nu}(\|K^{\varnothing }_{j}\|^{2}_{j}+\|V^{ \varnothing }_{j}\|^{2}_{j}))\).
Next we explain how to estimate \(e^{-V_{j+1}(U\setminus X)+u_{j}|X|}\pi _{\star }F\). The prefactor is at most \(2^{|\mathcal {B} _{j}(X)|}\), i.e., the analogue of Lemma 3.18 applies when \(V^{\varnothing }\), \(u^{\varnothing}\) and \(K^{\varnothing }\) are replaced by \(V\), \(u\) and \(K\) if \(\varepsilon \) is small enough, and it suffices to estimate \(\pi _{\star }F\).
Consider the small set contributions to \(\mathcal {L} _{j+1}(U)\), i.e., the analogue of Sect. 3.8.3. As stated previously, Lemma 3.15 is replaced with Lemma 5.12 whereas Lemmas 5.11 and 5.10 replace Lemmas 3.16 and 3.17. In detail, in the analogue of (3.78) we now also include quadratic terms in \(\delta I\), i.e., we replace (3.78) by
with the corresponding analogue of (3.77) then being (for \(X\in \mathcal {S} _{j}\setminus \mathcal {B} _{j}\))
Let us note that since \(B \cup B'\) is not necessarily connected (so in that case not a small set), along with the third term in (5.69), there is a corresponding correction for polymers in the large set sum (3.95): the terms inside the sum are replaced by \(e^{-V_{j+1}(U \setminus X)+u_{j+1}|X|}\) multiplied by
Now Lemma 5.12 bounds the sum over \(X\) of the \((1-\operatorname{Loc}_{X})\) terms in (5.68) and (5.69). Lemma 5.11 bounds the second term on the right-hand side of (5.68). Finally, (5.50) of Lemma 5.10 bounds the second term in (5.69) by \(O(\|V^{\varnothing }_{j}\|_{j}+\|K^{\varnothing }_{j}\|_{j})(\|V_{j} \|_{j}+L^{2(d-2)}\|K_{j}\|_{j})\), after making use of the cancellation between \((\delta I)^{X}\) and \(\delta I(B)\delta I(B')\) when \(X = B \cup B'\) and \(B' \subset \bar{B}\). Indeed, note that for all other \(X\) at least one \(B \in \mathcal {B} _{j}(X)\) does not contain \(a\) or \(b\). Putting these bounds together (as in the proof of Theorem 3.13) then gives that the small set contribution to \(\pi _{\star }\mathcal {L} _{j+1}(U)\) is \(O(L^{-(\frac{d-2}{2} \wedge 1)}\|K^{\star }_{j}\|_{j}) + O(\|V^{ \varnothing }_{j}\|_{j}+\|K^{\varnothing }_{j}\|_{j})(\|V_{j}\|_{j}+L^{2(d-2)} \|K_{j}\|_{j})\).
To bound the large set term (5.70) and the non-linear contributions, we will use the principle that for \(F_{i}\in \mathcal {V} \),
as the product of any three elements of \(\mathcal {V} ^{\star }\) is zero. The bound on the sum over
proceeds exactly as in Sect. 3.8.4, bearing in mind (5.71) and (5.28). The resulting estimate is \(O(A^{-\eta}\|K^{\star }_{j}\|_{j})\). For the second term in (5.70), observe that if \(|\mathcal {B} _{j}(X)|=2\) and \(\bar{X}\in \mathcal {B}_{j+1}\), the bound is identical to that of the same term in (5.69) above. The remaining possibilities are that either \(|\mathcal {B} _{j}(X)| \geqslant 3\) or \(|\mathcal {B} _{j}(X)|=2\) but with constituent \(j\)-blocks which are in distinct \((j+1)\)-blocks. In the former case, by applying (5.71), (5.50) of Lemma 5.10 and Lemma 3.17 and then proceeding as in Sect. 3.8.4, we obtain
The remaining case is \(|\mathcal {B} _{j+1}(U)|=2\) and \(|\mathcal {B} _{j}(X)|=2\) where \(U = \bar{X}\). Then the \(\delta I(B)\delta I(B'){\mathbf{1}}_{B \cup B'=X}\) cancellation is absent, but \(\pi _{\star }\mathbb{E}_{C_{j+1}}(\delta I)^{X}\) itself satisfies the desired bound by Lemma 5.10. Indeed, either \(a\) and \(b\) are in the same \((j+1)\) block or they are not. If they are, we use (5.50) with \(n=1\), and if not, this follows from (5.52) since \(a\) and \(b\) being in distinct \((j+1)\)-blocks of \(U\) implies that \(|a-b|_{\infty }\geqslant \frac{1}{2} L^{j+1}\) since \(a\) is positioned in the center of all of its blocks. The bound
follows as there are at most \(L^{2d}\) summands.
All together, after possibly increasing \(A\), we obtain that the large set contribution to \(\mathcal {L} _{j+1}(U)\) is
The non-linear contribution does not require any changes as the bound from Sect. 3.8.5 already gives (after possibly increasing \(A\)) \(A^{\nu }O(\|K_{j}\|_{j}(\|V_{j}\|_{j}+\|K_{j}\|_{j}))\). □
5.7 Flow of observable coupling constants
With Theorem 5.8 in place, the evolution of the observable coupling constants in \(u^{\star }+V^{\star }\) is the same as the free one from Sect. 5.2 up to the addition of remainder terms from the \(K\) coordinate. To avoid carrying an unimportant factor of \(L^{2(d-2)}\) through equations, recall that we write \(O_{L}(\cdot )\) to indicate bounds with constants possibly depending on \(L\) (but we reemphasise that implicit constants are always independent of the scale \(j\)).
Lemma 5.13
Suppose \(j< N\), \(x\in \{a,b\}\), and that (5.40) holds. If \(j< j_{ab}\), further suppose that \(K^{ab}_{j}(X)=0\) for \(X\in \mathcal {S} _{j}\). In Case (1),
and in Case (2),
Moreover, for \(j+1< N\), all coupling constants are independent of \(N\).
Note that there is no error term in the equation for \(\eta \), as the corresponding nonlocal field monomials \(\bar{\psi}_{a}\psi _{b}\) and \(\bar{\psi}_{b}\psi _{a}\) are not contained in the image of \(\operatorname{Loc}\).
Proof
For \(j< N\), the main contribution in (5.40) is identical to that in Lemma 5.3. The indicator functions \(1_{j\geqslant j_{ab}}\) in the error terms are due to the assumption \(K_{j}^{ab}(X)=0\) for \(j< j_{ab}\) and \(X\in \mathcal {S} _{j}\). The bounds for the error terms follow from the definition of the norms as in obtaining (4.10). Finally, that the couplings are independent of \(N\) is a consequence of the consistency of the renormalisation group map, i.e., Proposition 3.12 (applied to the renormalisation group map extended by observables). □
The next lemma shows that if we maintain control of \(\|K_{k}^{\star }\|_{k}\) up to scale \(j\) then we control the coupling constants in \(V^{\star }\) on scale \(j\).
Lemma 5.14
Assume that \(\|K^{\star }_{k}\|_{k} \leqslant M \lambda _{0}b_{0}L^{-\kappa k}\) for \(k< j\) and that (5.40) holds for \(k< j\). Then, in Case (1) if \(q_{0}=r_{0}=0\) and \(\lambda _{0}>0\),
and, in Case (2), if \(q_{0}=r_{0}=\gamma _{x,0}=\eta _{0}=0\) and \(\lambda _{0}>0\),
where \(\lambda _{j}=\lambda _{x,j}\) for either \(x=a\) or \(x=b\). In both Cases (1) and (2),
Proof
The bounds on the coupling constants in (5.84)–(5.88) follow from Lemma 5.13; the hypothesis regarding \(K_{j}(X)=0\) for \(j< j_{ab}\) and \(X\in \mathcal {S} _{j}\) holds as Definition 5.7 implies that for an iteration \((V_{j},K_{j})\) of the renormalisation group map, the \(\mathcal {N} ^{ab}\) components of \(V_{j}(B)\) and \(K_{j}(X)\) with \(X\in \mathcal {S} _{j}\) can only be nonzero for \(j>j_{ab}\) since we have started the flow with \(r_{0}=0\) in Case (1), and \(q_{0}=\eta _{0}=r_{0}=0\) in Case (2). What remains is to analyse the recurrences to establish (5.89).
For \(\lambda _{x,j}\), since \(\ell _{x,j}^{-1}\ell _{j}^{-p} = 1\) in Case (\(p\)), using (5.76), respectively (5.79),
The bounds on \(r_{j}\) follow from the fact that all contributions are 0 for scales \(j< j_{ab}\) if \(r_{0}=0\). For example, in Case (2), using (5.83),
Case (1) is similar, except no factor \(\ell _{j_{ab}}\) arises (see (5.25)). The bound on \(\eta _{j}\) in Case (2) follows from the preceding analysis of \(\lambda _{x,j}\), the fact that \(\eta _{j}=0\) for \(j< j_{ab}\) if \(\eta _{0}=0\) since \(C_{j}\) has finite range (\(C_{j}(a,b)=0\) if \(|a-b|_{\infty }\geqslant \frac{1}{2} L^{j}\)), and that \(C_{j+1}(a,b) \leqslant O_{L}(L^{-(d-2)j})\):
For the bound on the norm of \(\|V^{\star }_{j}\|_{j}\) recall that the \(q\) and \(\gamma \) terms have been taken out of \(V^{\star }\). Thus in Case (1),
Similarly, in Case (2), using that \(\ell _{j}^{2}\ell _{ab,j}= \ell _{j\wedge j_{ab}}^{-2} = O_{L}(|a-b|^{d-2})\) for \(j \geqslant j_{ab}\),
□
We now analyse the observable flow from initial conditions which extend the bulk initial conditions of Theorem 3.24, and verify the assumption on \(K^{\star }\) made in Lemma 5.14 along this flow. As already remarked at the beginning of Sect. 5.5, the renormalisation group maps with observables are upper triangular, so that the observable components of the maps do not affect the bulk flow. Thus from now on, the bulk components \((V_{j}^{\varnothing},K_{j}^{\varnothing})_{j \leqslant N}\) are identified with the trajectory given by Theorem 3.24 and we may use the decay rates from that theorem as inputs in obtaining estimates on the remaining components. In particular, there is an \(\alpha >0\) such that
Using this as input, we iterate the observable flow (5.40)–(5.41), with initial condition \(\lambda _{a,0}=\lambda _{b,0}=\lambda _{0}\) small enough and all other observable coupling constants equal to 0.
Proposition 5.15
Assume that the bulk renormalisation group flow \((V^{\varnothing }_{j},K^{\varnothing }_{j})\) obeys
for some \(\alpha >0\). Then there is \(\kappa >0\) such that for \(\lambda _{a,0}=\lambda _{b,0}=\lambda _{0} >0\) and \(b_{0}>0\) sufficiently small and all other observable coupling constants initially 0,
Proof
We may assume \(\lambda _{0}<1\), and that \(\kappa \) is less than \(\alpha \) and the exponents of \(L\) and \(A\) in (5.41). The proof is by induction. The inductive assumption is that \(\|K^{\star }_{k}\|_{k} \leqslant M b_{0}\lambda _{0}L^{-\kappa k}\) for all \(k \leqslant j\), for some \(M=M(L)\) chosen large enough below. Clearly, this holds for \(j=0\). Lemma 5.14 then shows that \(\|V^{\star }_{k}\|_{k}\leqslant O_{L}(1)\lambda _{0} + O_{L}(1) M \lambda _{0}b_{0}\) for all \(k\leqslant j+1\). We now apply Theorem 5.8 to control \(K^{\star }_{j+1}\). Since \(A\) is chosen as function of \(L\), the second term on the right-hand side of (5.41) is
as long as \(b_{0}\leqslant \lambda _{0}\) and \(Mb_{0} + \lambda _{0}\) is sufficiently small (depending on \(L\)). As \(A>L\), and using our second assumption on \(\kappa \), we obtain from (5.41) that
provided that \(M\) is sufficiently large and \(b_{0}\) is sufficiently small (depending only on \(L\)). Hence if \(\lambda _{0}\) and \(b_{0}\) are small enough we have advanced the induction, completing the proof. □
6 Computation of pointwise correlation functions
In this section we use the results of Sect. 5 to prove the following estimates for the pointwise correlation functions \(\langle \bar{\psi}_{a}\psi _{b} \rangle \), \(\langle \bar{\psi}_{a}\psi _{a} \rangle \), and \(\langle \bar{\psi}_{a}\psi _{a}\bar{\psi}_{b}\psi _{b} \rangle \). Recall (2.27) and (2.25), i.e.,
Thus \(W_{N}(x)\) is essentially the torus Green function \((-\Delta +m^{2})^{-1}(0,x)\) with the zero mode subtracted and \(t_{N}\) is essentially \(m^{-2}\) when \(L^{2N} \ll m^{-2}\) and negligible otherwise.
Proposition 6.1
For \(b_{0}\) sufficiently small and \(m^{2} \geqslant 0\), there exists continuous functions
such that if \(V^{\varnothing }_{0}=V_{0}^{c}(m^{2},b_{0})\) is as in Theorem 3.24, \(V^{\star }_{0}\) is as in Proposition 5.15, and \(\tilde{u}_{N,N}^{c}=\tilde{u}_{N,N}^{c}(b_{0},m^{2})\) is as in Proposition 4.1, then
Proposition 6.2
Under the same assumptions as in Proposition 6.1,
Throughout this section, we assume that the renormalisation group flow \((V_{j},K_{j})_{j \leqslant N}\) is given as in Corollary 3.26 (bulk) and Proposition 5.15 (observables).
6.1 Integration of the zero mode
As in the analysis of the susceptibility in Sect. 4, we treat the final integration over the zero mode explicitly. Again we will only require the restriction to constant \(\psi \), \(\bar{\psi}\) (as discussed below (4.3)) of
where the last equation defines \(\tilde{Z}_{N,N}\). We write \(\tilde{Z}_{N,N}=\tilde{Z}_{N,N}^{\varnothing}+\tilde{Z}_{N,N}^{\star }\) for its decomposition into bulk and observable parts (see (5.23)). The bulk term was already computed in Proposition 4.1. The observable term \(\tilde{Z}_{N,N}^{\star }\) is computed by the next lemma. In the lemma we only give explicit formulas for the terms that will be used in the proofs of Propositions 6.1 and 6.2.
Lemma 6.3
Restricted to constant \(\psi \), \(\bar{\psi}\), in Case (1),
where
In Case (2),
where, setting \(\tilde{\lambda}_{x,N,N} = \lambda _{x,N} +O_{L}(\ell _{x,N}^{-1} \ell _{N}^{-2} \|K^{\star }_{N}\|_{N})\),
The error bounds above reveal the tension in the explicit choices of \(\ell _{x,j}^{-1}\) and \(\ell _{ab,j}^{-1}\). To obtain effective error estimates, we want \(\ell _{x,N}^{-1}\) and \(\ell _{ab,N}^{-1}\) to be as small as possible. On the other hand, to control the iterative estimates of Theorem 5.8 over the entire trajectory, i.e., to prove Proposition 5.15, we needed that \(\ell _{x,j}\) and \(\ell _{ab,j}\) were not too large. In particular, either of the more naive choices \(\ell _{ab,j}=\ell _{j}^{-4}\) and \(\ell _{ab,j} = \ell _{j_{ab}}^{-4}\) in Case (2) would have lead to difficulties, either in terms of forcing us to track additional terms in the flow and in terms of controlling norms inductively, or by leading to error bounds that are not strong enough to capture the zero mode sufficiently accurately.
Proof
Throughout the proof, we restrict to constant \(\psi \), \(\bar{\psi}\). Since
by applying \(\mathbb{E}_{t_{N}Q_{N}}\theta \) we obtain
In obtaining \(A\) we used (4.4) which gives \(\mathbb{E}_{t_{N}Q_{N}}[\theta (e^{-V_{N}^{\varnothing}(\Lambda )}+K_{N}^{ \varnothing}(\Lambda ))]= 1+\tilde{u}_{N,N}-|\Lambda _{N}|\tilde{a}_{N,N} \psi \bar{\psi}\). Since each term in \(V_{N}^{\varnothing}(\Lambda )\) contains a factor \(\psi \bar{\psi}\) and each term in \(V_{N}^{\star }(\Lambda )\) either \(\psi \) or \(\bar{\psi}\), we have \(V_{N}^{\varnothing}(\Lambda )V_{N}^{\star }(\Lambda )=0\). Thus
Case (1). Since \(\sigma _{a}^{2}=\bar{\sigma}_{b}^{2}=0\),
we get
The constants \(k_{N}^{\#}\) are given in terms of derivatives of \(K_{N}(\Lambda )\) and bounded analogously as in (4.10). For example, \(k_{N}^{\sigma _{a}\bar{\psi}} = O_{L}(\ell _{a,N}^{-1}\ell _{N}^{-1}\|K^{ \star }_{N}\|_{N})\), and similarly for the other \(k_{N}^{\#}\) terms, the rule being that we have a factor \(\ell _{x,N}^{-1}\) if there is a superscript \(\sigma _{a}\) or \(\bar{\sigma}_{b}\) but not both, a factor \(\ell _{ab,N}^{-1}\) for \(\sigma _{a}\bar{\sigma}_{b}\) and a factor \(\ell _{N}^{-1}\) for each superscript \(\psi \) or \(\bar{\psi}\). These bounds follow from the definition of the \(T_{j}(\ell _{j})\) norm.
Since \(\mathbb{E}_{t_{N}Q_{N}}\theta \psi \bar{\psi}= -t_{N}|\Lambda _{N}|^{-1}+ \psi \bar{\psi}\) the claim follows by collecting terms and using (3.7).
Case (2). Using again that \(\sigma _{a}^{2}=\sigma _{b}^{2}=0\), but now taking in account that \(u^{\star }(\Lambda )\) has additional terms compared to Case (1),
and therefore
Since in Case (2) each term in \(V^{\star }_{N}(\Lambda )\) contains a factor of \(\psi \bar{\psi}\), we have \(V^{\star }_{N}(\Lambda )^{2}=0\) and thus
Therefore
where we have set \(\tilde{\lambda}_{x,N} = \lambda _{x,N} + k_{N}^{\sigma _{x}\bar{\psi}\psi}\). Taking the expectation and collecting all terms gives
The bounds on the constants \(k_{N}^{\#}\) are analogous to those in Case (1). □
6.2 Analysis of one-point functions
We now analyse the observable flow given by Lemma 5.13 to derive the asymptotics of the correlation functions. Note that the coupling constants \(\lambda _{x,j}\) and \(\gamma _{x,j}\) can possibly depend on \(x=a,b\) as the contributions from \(K\) can depend on the relative position of the points in the division of \(\Lambda _{N}\) into blocks. The following lemma shows that in the limit \(j\to \infty \) they become independent of \(x\); an analogous argument was used in [14, Lemma 4.6].
Lemma 6.4
Under the hypotheses of Proposition 6.1there are \(\lambda _{\infty}^{(p)} = \lambda _{0} + O_{L}(\lambda _{0}b_{0})\) and \(\gamma _{\infty}= O_{L}(\lambda _{0})\), all continuous in \(m^{2}\geqslant 0\) and \(b_{0}\) small, such that for \(x\in \{a,b\}\),
In Case (1), \(\lambda _{\infty}^{(1)} = \lambda _{0}\). In Case (2), \(\lambda _{\infty}^{(2)} = \lambda _{0} + O_{L}(\lambda _{0}b_{0})\) and \(\gamma _{\infty}^{(2)} = \lambda _{\infty}^{(2)}(-\Delta ^{ \mathbb{Z}^{d}}+m^{2})^{-1}(0,0)+ O_{L}(\lambda _{0}b_{0})\), and with the abbreviations \(\lambda =\lambda ^{(2)}\) and \(\gamma =\gamma ^{(2)}\),
Proof
We will typically drop the superscript \((p)\). In both cases, we have already seen that
Since the \(K^{\star }_{k}\) are independent of \(N\) for \(k< N\) (by Proposition 3.12 for the extended renormalisation group map, see Sect. 5.5), the limit \(\lambda _{x,\infty}\) makes sense, exists, and \(|\lambda _{x,j}-\lambda _{x,\infty}| = O_{L}(\lambda _{0}b_{0}L^{- \kappa j})\). Similarly, in Case (2), by Lemma 5.13 and \(\ell _{x,j}^{-1} = \ell _{j}^{2} =O_{L}(L^{-(d-2)j})\),
In particular, by the above estimate for \(|\lambda _{x,j}-\lambda _{x,\infty}|\), we have
The continuity claims follow from the continuity of the covariances \(C_{j}\) in \(m^{2}\geqslant 0\), of the renormalisation group coordinates \(K_{j}\), and that both \(\lambda _{\infty}\) and \(\gamma _{\infty}\) are uniformly convergent sums of terms continuous in \(b_{0}\) and \(m^{2}\geqslant 0\).
To show that \(\lambda _{x,\infty}^{(1)}=\lambda _{0}\) in Case (1), which is in particular independent of \(x\), we argue as in the proof of [14, Lemma 4.6]. On the one hand, Lemma 6.3 implies as \(N\to \infty \) with \(m^{2}>0\) fixed,
where \(|_{0}\) denotes projection onto the degree 0 part, i.e., \(\psi =\bar{\psi}=\sigma =\bar{\sigma}=0\), and we have dropped the superscript (1) from \(\lambda _{a,j}\). On the other hand, we claim
Indeed, the first equality in (6.32) follows analogously to [14, (4.51)–(4.53)]: let \(\Gamma (\rho ,\bar{\rho})\) be as in (4.12), except that \(Z_{0}\) now includes the observable terms \(\sigma _{a}\) and \(\bar{\sigma}_{b}\) and we write \(\rho \) and \(\bar{\rho}\) for the constant external field to distinguish them from \(\sigma _{a}\) and \(\bar{\sigma}_{b}\). Then as in (4.15),
and \(\langle \cdot \rangle _{\sigma _{a},\bar{\sigma}_{b}}\) denotes the expectation that still depends on the source fields \(\sigma _{a}\) and \(\bar{\sigma}_{b}\). Differentiating with respect to \(\sigma _{a}\) and setting \(\bar{\sigma}_{b}=0\) gives
which is the first equality of (6.32) upon rearranging. The second equality in (6.32) follows from Proposition 4.2.
The right-hand side of (6.32) converges to \(\lambda _{0}\) in the limit \(N\to \infty \) with \(m^{2}>0\) fixed since \(\tilde{a}_{N,N}= a_{N} - k_{N}^{2}/|\Lambda _{N}| = O_{L}(L^{-2N}\|V_{N} \|_{N})+O_{L}(L^{-2N}\|K_{N}\|_{N}) \to 0\) and \(\tilde{u}_{N,N}= k_{N}^{0} + \tilde{a}_{N,N}t_{N} = O(\|K_{N}\|_{N}) + \tilde{a}_{N,N}t_{N} \to 0\) when \(m^{2}>0\) is fixed. Since the left-hand sides of (6.31)–(6.32) are equal, we conclude that \(\lambda _{a,\infty}=\lambda _{0}\) when \(m^{2}>0\). By continuity this identity then extends to \(m^{2}=0\).
In Case (2), to show (6.27), we use (6.12), that Proposition 5.15 implies \(\|K^{\star }_{N}\|_{N} = O_{L}(\lambda _{0}b_{0}L^{-\kappa N})\), and \(\ell _{x,N}^{-1}\ell _{N}^{-2} = 1\) and \(\ell _{x,N}^{-1}= \ell _{N}^{2} = O_{L}(L^{-(d-2)N})\) to obtain
Since
this gives (6.27). In particular, by the translation invariance of \(\langle \bar{\psi}_{a}\psi _{a} \rangle \), taking \(N\to \infty \) with \(m^{2}>0\) fixed implies \(\gamma _{a,\infty}\) is independent of \(a\). Similarly, taking \(m^{2}\downarrow 0\) first and then \(N\to \infty \) we see that \(\lambda _{a,\infty}\) is independent of \(a\). Indeed, using (4.4), as \(N\to \infty \),
where \(\langle \bar{\psi}_{a}\psi _{a} \rangle \) and all scale-dependent coupling constants are evaluated at \(m^{2}=0\). Thus \(\lambda _{a,\infty} =\lim _{N\to \infty} |\Lambda _{N}|\tilde{a}_{N,N} (\lambda _{0} \langle \bar{\psi}_{a}\psi _{a} \rangle -\gamma _{a, \infty})\) and the right-hand side is independent of \(a\). □
Proof of Proposition 6.1
Taking \(\lambda _{0}>0\) small enough, the proposition follows immediately from Lemma 6.4 with \(\lambda = \lambda _{\infty}^{(2)}/\lambda _{0}\) and \(\gamma = \gamma _{\infty}^{(2)}/\lambda _{0}\). □
6.3 Analysis of two-point functions
Next we derive estimates for the two-point functions.
Lemma 6.5
Under the hypotheses of Proposition 6.1,
and, setting \(\lambda _{\infty }= \lambda ^{(2)}_{\infty}\) and \(\gamma _{\infty }= \gamma ^{(2)}_{\infty}\) as in Lemma 6.4,
Proof
The proofs of (6.38) and (6.39) corresponding to Cases (1) and (2) are again analogous.
Case (1). By Lemma 5.14 (whose hypotheses are verified by Proposition 5.15) and Lemma 6.4,
Using that \(\ell _{ab,j}^{-1}\|K^{ab}_{j}\|_{j} \leqslant O_{L}(\lambda _{0}b_{0}L^{-(d-2+ \kappa )j})1_{j\geqslant j_{ab}}\) and \(|C_{j+1}(a,b)| \leqslant C_{j+1}(0,0) \leqslant O_{L}(L^{-(d-2)j})\) it then follows from Lemma 5.13 that
where we have used (6.40), \(|a-b|\leqslant L\), that \(C_{j}(a,b)=0\) for \(j< j_{ab}\), and that \(W_{N}(x-y)=C_{1}(x,y) + \cdots + C_{N}(x,y)\). By (6.10), using that \(\ell _{ab,N}^{-1}\ell _{N}^{-2}=1\) and again (6.40), therefore
Since \(\langle \bar{\psi}_{a}\psi _{b} \rangle = \tilde{Z}_{N,N}^{ \bar{\sigma}_{b}\sigma _{a}}/(\lambda _{0}^{2} (1+\tilde{u}_{N,N}))\) and \(|\lambda _{0}| \leqslant 1\), the claim for the two-point function follows.
Case (2). Again, the analogue of (6.40) holds:
The first estimate is by Lemma 6.4, the second by Lemma 5.14. Since \(C_{k}(a,b)=0\) for \(k< j_{ab}\) and \(|C_{k+1}(a,b)| \leqslant O_{L}(L^{-(d-2)k})\), then by Lemma 5.13 and as \(|a-b|\leqslant L\),
Note that
As a result, again by Lemma 5.13, these bounds together then give
We finally substitute these estimates into (6.13). Using also that \(\ell _{ab,N}^{-1}\ell _{N}^{-2} = \ell _{j_{ab}}^{2} = O_{L}(|a-b|^{-(d-2)})\), that \(\ell _{x,N}^{-1}\ell _{N}^{-2} = O_{L}(1)\), that \(\gamma _{x,N}=\gamma _{\infty}+O_{L}(b_{0}\lambda _{0}L^{-(d-2+ \kappa )N})\) by (6.26), and \(\|K^{\star }_{N}\|_{N} \leqslant O_{L}(b_{0}\lambda _{0}L^{-\kappa})\), we obtain
which gives (6.39) since \(\langle \bar{\psi}_{a}\psi _{a}\bar{\psi}_{b}\psi _{b} \rangle = \tilde{Z}_{N,N}^{\sigma _{a}\sigma _{b}}/(\lambda _{0}^{2}(1+\tilde{u}_{N,N}))\). □
Proof of Proposition 6.2
The proposition follows immediately from Lemma 6.5 with the same \(\lambda \) and \(\gamma \) as in Proposition 6.1. □
7 Proof of Theorems 2.1 and 2.3
Proof of Theorems 2.1 and 2.3
By summation by parts on the whole torus \(\Lambda _{N}\), we have
Given \(m^{2}\geqslant 0\) and \(b_{0}\) small, we choose \(V_{0}^{c}(b_{0},m^{2})\) as in Theorem 3.24. This defines the functions \(s_{0}^{c} = y_{0}^{c}+z_{0}^{c}\) and \(a_{0}^{c}\) in (2.5) with the required regularity properties. The claims for the correlation functions and the partition function then follow from Propositions 4.1–4.2 and 6.1–6.2. The continuity of \(u_{N}^{c}\) follows from the continuity of \(V_{0}^{c}\) and the continuity of the renormalisation group maps.
For Theorem 2.1, note that the statements simplify by the assumption \(m^{2} \geqslant L^{-2N}\). Indeed, using that \((m^{2}|\Lambda _{N}|)^{-1} \leqslant L^{-(d-2)N}\) and \(|a_{N}| \leqslant O_{L}(b_{0}L^{-(2+\kappa )N})\), by Proposition 4.1, we have that \(|\tilde{a}_{N,N}|\leqslant O_{L}(b_{0}L^{-(2+\kappa )N})\) and \(|\tilde{u}_{N,N}| \leqslant O_{L}(b_{0}L^{-\kappa N})\). □
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Acknowledgements
We thank David Brydges and Gordon Slade. This article would not have been possible in this form without their previous contributions to the renormalisation group method. We also thank them for their permission to include Fig. 1 from [18]. We thank the referees for their helpful comments.
R.B. was supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 851682 SPINRG). N.C. was supported by Israel Science Foundation grant number 1692/17.
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Appendices
Appendix A: Random forests and the \(\mathbb{H}^{0|2}\) model
1.1 A.1 Proof of Proposition 1.4
For any graph \(G=(\Lambda ,E)\) with edge weights \((\beta _{xy})\) and vertex weights \((h_{x})\), the partition function appearing in (1.1) can be generalised to
where ℱ is the set of forest subgraphs of \(G\). Recall from the discussion above (1.3) that expanding the product over \(T\) in (A.1) can be interpreted as choosing, for each \(T\), either (i) a root vertex \(x\in T\) with weight \(h_{x}\) or (ii) leaving \(T\) unrooted. This interpretation will be used in Lemma A.4.
By [20, Theorem 2.1] (which follows [37]),
Moreover, by [20, Corollary 2.2], if \(h=0\) then
Proposition 1.4 follows easily from this. For convenience, we restate the proposition as follows. In the statement and throughout this appendix, inequalities like \(\beta \geqslant 0\) are to be interpreted pointwise, i.e., \(\beta _{xy}\geqslant 0\) for all edges \(xy\).
Proposition A.1
For any finite graph \(G\), any \(\beta \geqslant 0\) and \(h \geqslant 0\),
and the normalising constants in (1.1) and (1.13) are equal. In particular,
Proof of Proposition A.1
For notational ease, we write the proof for constant \(h\). The equality of the normalising constants is a special case of (A.2). To see (A.4), we use that \((z_{0}-1)^{2}=0\) so that \(z_{0}=1-(1-z_{0})=e^{-(1-z_{0})}\). As a result \(\langle z_{0} \rangle _{\beta ,h} = Z_{\beta ,h-1_{0}} / Z_{\beta ,h}\), and (A.1) gives
Similarly, \(\langle z_{0}z_{x} \rangle = Z_{\beta ,h-1_{0}-1_{x}}/Z_{\beta ,h}\) and thus (A.1) shows that
To see (A.6), we note that the left-hand side is the connection probability in the extended graph \(G^{\mathfrak{g}}\). From (A.3) with \(\beta _{xy} = \beta \) for \(x,y\in \Lambda \) and \(\beta _{x\mathfrak{g}} = h\) for \(x\in \Lambda \) we thus obtain the claim:
To see (A.5), we combine (A.9) and (A.10) to get
Finally, (A.7) is (A.3). This completes the proof. □
The extended graph \(G^{\mathfrak{g}}\) allows \(z\)-observables to be interpreted in terms of edges connecting vertices in the base graph \(G\) to \(\mathfrak{g}\). To state this, we denote by \(\{x\mathfrak{g}\}\) the event the edge between \(x\) and \(\mathfrak{g}\) is present. The next lemma will be used in Appendix A.3.
Proposition A.2
Proof
As discussed above, after expanding the product in (A.1) the external fields \(h_{x}\) can be viewed as edge weights for edges from \(x\) to \(\mathfrak{g}\). With this in mind the formulas follow by differentiating (A.2). □
1.2 A.2 High-temperature phase and positive external field
Proposition A.3
If \(\beta < p_{c}(d)/(1-p_{c}(d))\), then \(\theta _{d}(\beta )=0\). Moreover, there is a \(c=c(\beta )>0\) such that \(\mathbb{P}^{\Lambda _{N}}_{\beta ,0}[0\leftrightarrow x] \leqslant e^{-c|x|}\).
Proof
In finite volume, Holley’s inequality implies the stochastic domination \(\mathbb{P}^{\Lambda _{N}}_{\beta ,h}\preceq \mathbb{P}^{\Lambda _{N}}_{p,r}\), where the latter measure is Bernoulli bond percolation on the extended graph \(G^{\mathfrak{g}}\) with \(p=\beta /(1+\beta )\) and \(r=h/(1+h)\), see [20, Appendix A]. In particular,
Since each edge to the ghost is chosen independently with probability \(r\), this latter quantity is
since \(1-(1-r)^{n}\leqslant rn\) for \(0\leqslant r\leqslant 1\). Here \(C_{0}\) is the cluster of the origin on the torus without the ghost site, so \(\mathbb{E}^{\Lambda _{N}}_{p,r}\lvert C_{0} \rvert =\mathbb{E}^{ \Lambda _{N}}_{p,0}\lvert C_{0} \rvert \). Now suppose \(\beta \) is such that \(p< p_{c}(d)\). Then the right-hand side is finite and uniformly bounded in \(N\). Hence
The second claim follows from stochastic domination, as when \(p< p_{c}(d)\) bond percolation has exponentially decaying connection probabilities [56]. □
Lemma A.4
Let \(h>0\) and suppose that for all \(x\), \(h_{x}= h\). Then there are \(c,C>0\) depending on \(d\), \(\beta \), \(h\) such that
Proof
We begin with the inequality on the left of (A.17). Define \(\mathcal {F} (0\leftrightarrow x)\) to be the set of forests in which both 0 is connected to \(x\) and \(T_{0}\) is rooted at 0, and ℱ the set of all forests. In this argument we treat ℱ as being a set of (possibly) rooted forests, i.e., we identify edges to \(\mathfrak{g}\) with roots. Without loss of generality we may assume \(x\cdot e_{1} \geqslant \alpha |x|\) for a fixed \(\alpha >0\). Note that if \(F\in \mathcal {F} (0\leftrightarrow x)\) there is a unique path \(\gamma _{F}\) from 0 to \(x\) in \(F\), and there are at least \(\alpha |x|\) edges of the form \(\{u,u+e_{1}\}\) in \(\gamma _{F}\).
We define a map \(S\colon \mathcal {F} (0\leftrightarrow x) \to 2^{ \mathcal {F} }\) by, for \(F\in \mathcal {F} (0\leftrightarrow x)\),
-
1.
choosing a subset \(\{u_{i},v_{i}\}\) of the edges \(\{ \{u,v\} \in \gamma _{F} \mid v=u+e_{1}\}\), and
-
2.
removing each \(\{u_{i},v_{i}\}\) and rooting the tree containing \(v_{i}\) at \(v_{i}\).
Thus \(S(F)\) is the set of forests that results from all possible choices in the first step. The second step does yield an element of \(2^{\mathcal {F} }\) since \(T_{0}\) is rooted at 0, so it cannot be the case that the tree containing \(v_{i}\) is already rooted (connected to \(\mathfrak{g}\)).
The map \(S\) is injective, meaning that given \(\bar{F}\in \bigcup _{F\in \mathcal {F} (0\leftrightarrow x)}S(F)\) there is a unique \(F\) such that \(\bar{F} \in S(F)\). Indeed, given \(\bar{F}\in S(F)\), \(F\) can be reconstructed as follows. In \(\bar{F}\), either the tree containing \(x\) contains 0, or else it is rooted at a unique vertex \(v'\) and it is not connected to \(u'=v'-e_{1}\). Set \(\bar{F}'=\bar{F}\cup \{u',v'\}\). The previous sentence applies to \(\bar{F}'\) as well, and continuing until a connection to 0 is formed we recover \(F\). This reconstruction was independent of \(F\), and hence if \(\bar{F}_{1}=\bar{F}_{2}\), \(\bar{F}_{i}\in S(F_{i})\), we have \(F_{1}=F_{2}\).
Let \(w(F)= h \beta ^{F}\prod _{T\neq T_{0}}(1+h|V(T)|)\). Then for \(\bar{F} \in S(F)\), \(w(\bar{F}) = w(F) (\frac{h}{\beta})^{k}\) if \(\bar{F}\) had \(k\) edges removed. Hence if the connection from 0 to \(x\) in \(F\) has \(k\) edges of the form \(\{u,v\}\), \(v=u+e_{1}\),
Let \(\mathcal {F} _{k}(x) \subset \mathcal {F} (0 \leftrightarrow x)\) be the set of forests where the connection from 0 to \(x\) contains \(k\) edges of the form \(\{u,v\}\), \(v=u+e_{1}\). We have the lower bound
since \(S\) is injective and all of the summands are non-negative. Hence we obtain, using (A.18),
A similar argument applies when \(0\nleftrightarrow \mathfrak{g}\); this condition is used in the second step defining \(S\) to ensure the trees containing the vertices \(v_{i}\) are not already connected to \(\mathfrak{g}\). In this case the weight \(w(F)\) does not have the factor \(h\), but the remainder of the argument is identical. □
Proposition A.5
Let \(h>0\) and suppose that for all \(x\), \(h_{x}= h\). Then there are \(c,C>0\) depending on \(d\), \(\beta \), \(h\) such that
Proof
Since
it is enough to estimate the first term, as the second is covered by Lemma A.4. Note
where the first equality follows from the fact that the only one vertex per component may connect to \(\mathfrak {g}\), and the second follows from exchangeability of the choice of root. Examining the rightmost expression, there are most \(c_{d} |x |^{d}\) summands in which \(|y |\leqslant |x|\); for these terms we drop the condition \(0\leftrightarrow y\). For the rest we drop \(0\leftrightarrow x\). This gives, by Lemma A.4,
where \(c\), \(C\) are changing from location to location but depend on \(d\), \(\beta \), \(h\) only. □
1.3 A.3 Infinite volume limit
We now discuss weak limits \(\mathbb{P}^{\mathbb{Z}^{d}}_{\beta}\) obtained by (i) first taking a (possibly subsequential) infinite-volume weak limit \(\mathbb{P}^{\mathbb{Z}^{d}}_{\beta ,h} = \lim _{N}\mathbb{P}^{ \Lambda _{N}}_{\beta ,h}\) and (ii) subsequently taking a (possibly subsequential) limit \(\mathbb{P}^{\mathbb{Z}^{d}}_{\beta}=\lim _{h\downarrow 0 }\mathbb{P}^{ \mathbb{Z}^{d}}_{\beta ,h}\). We do not explicitly indicate the convergent subsequence chosen as what follows applies to any fixed choice. Define
where the second equality is due to (A.12). Since this last display only involves cylinder events,
where the last equality defines \(\theta _{d}(\beta ,h)\).
Proposition A.6
Assume \(\lim _{h\downarrow 0}\theta _{d}(\beta ,h)=\theta _{d}(\beta )\) exists. Then
Proof
Write \(\mathbb{P}_{\beta ,h}=\mathbb{P}^{\mathbb{Z}^{d}}_{\beta ,h}\). We claim that
and hence, since \(\theta _{d}(\beta ,h) = 1-h^{-1}\mathbb{P}_{\beta ,h}[0\mathfrak{g}]\),
Granting the claim, by dominated convergence we obtain
as desired. To prove the claim, rewrite it as
The probabilities inside the limit are probabilities of cylinder events, and hence are limits of finite volume probabilities. For a fixed \(r\) the probability is at most \(h/(1+rh)\) in finite volume, which vanishes as \(r\to \infty \). □
Appendix B: Finite range decomposition
In this appendix, we give the precise references for the construction of the finite range decomposition (3.1). The general method we use was introduced in [12], and presented in the special case we use in [18, Chap. 3] and we will use this reference. For \(t>0\), first recall the polynomials \(P_{t}\) from [18, Chap. 3] (these polynomials are called \(W_{t}^{*}\) in [12]). These are polynomials of degree bounded by \(t\) satisfying
for any \(s>0\) and \(u \in [0,2]\). Our decomposition (3.1) is defined by
where \(\lambda (k) = 4\sum _{j=1}^{d}\sin ^{2}(k_{j}/2)\) and \(\Lambda _{N}^{*} \subset [-\pi ,\pi )^{d}\) is the dual torus. The estimates for \(C_{1}, \dots , C_{N-1}\) are straightforward from these Fourier representations and can be found in [18, Chap. 3]. We remark that in [18, Sect. 3.4], the torus covariances are defined by periodisation of the finite range covariances on \(\mathbb{Z}^{d}\); by Poisson summation this is equivalent to the above definition.
The decomposition of \(C_{N,N}\) in (3.6) is defined by removing the zero mode from \(C_{N,N}\):
from which (3.6) is immediate. For \(C_{N}\) estimates follows as in [18, Chap. 3]:
and analogously for the discrete gradients. Finally, by (B.1),
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Bauerschmidt, R., Crawford, N. & Helmuth, T. Percolation transition for random forests in \(d\geqslant 3\). Invent. math. (2024). https://doi.org/10.1007/s00222-024-01263-3
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DOI: https://doi.org/10.1007/s00222-024-01263-3