Loop Correlations in Random Wire Models
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Abstract
We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson–Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson–Dirichlet counterpart.
1 Introduction
Loop soups are models in statistical mechanics that involve sets of onedimensional loops living in higher dimensional space. These models are representations of particle or spin systems of statistical physics. It was recently conjectured that in most cases, in dimensions three and higher, these models have phases with long, macroscopic loops—the lengths of these loops scale like the volume of the system—and the joint distribution of macroscopic loops is always Poisson–Dirichlet [23].
This conjecture has been rigorously established in a model of spatial permutations related to the quantum Bose gas [10, 14, 18]. This model has a peculiar structure that makes it possible to integrate out the spatial variables and to use tools from asymptotic analysis, so there were suspicions that this property was accidental. But the conjecture has also been verified numerically in several other models, namely in lattice permutations [25]; in loop O(N) models [29]; and in the random interchange and a closely related loop model [2]. These findings are alas not supported by rigorous results.
There exist limited results for some models with fundamental spatial structure. The method of reflection positivity and infrared bounds [17, 21] allows to prove the occurrence of macroscopic loops [36]. More precisely, it is shown that the expectation of the length of a loop attached to a given site, when divided by the volume, is bounded away from 0 uniformly in the size of the system. While encouraging, this result gives no information regarding the possible presence of several macroscopic loops, let alone their joint distribution.
The goal of this article is to propose a genuinely spatial loop model where much of the conjecture can be rigorously established. We refer to it as the “random wire model”. It is defined for arbitrary finite graphs; in the most relevant case, the set of vertices is a large box in \({\mathbb Z}^d\) and the set of edges are the pairs of nearestneighbours. On each edge there is a random number of “links” satisfying the constraint that the number of links touching a site is even. These links are paired at each site, resulting in closed trajectories (an illustration can be found in Fig. 2). Our main result is a rigorous proof that even loop correlations are given by Poisson–Dirichlet, at least when the parameters of the model are chosen wisely.
There is a lot of background for this study. Our random wire model is an extension of the random current representation of the Ising model that was introduced by Griffiths, Hurst, and Sherman [24], and popularised by Aizenman [1]. It is also related to the BrydgesFröhlichSpencer representation of spin O(N) models [15, 19], and to loop O(N) models [9, 30]. The Poisson–Dirichlet distribution of random partitions was introduced by Kingman [27]; it is the invariant measure for the splitmerge (coagulationfragmentation) process [7, 16, 35]. Its relevance for meanfield loop soup models was first suggested by Aldous for the random interchange model on the complete graph, see [5]; Schramm succeeded in making this rigorous [34] (see also [6, 11, 12, 13]). The relevance of these ideas for systems with spatial structure was pointed out in [23].
The connections between the Poisson–Dirichlet distribution and symmetry breaking was noticed and exploited in [29, 36]. Our method of proof combines these ideas and rests on two major results about the classical XY model: The proof of Fröhlich, Simon, and Spencer that a phase transition occurs in dimensions three and higher [21]; and Pfister’s characterisation of all translationinvariant extremal infinitevolume Gibbs states [32]. We should point out that the precise relations between Poisson–Dirichlet and symmetry breaking are far from elucidated. The heuristics of Sect. 4.2 show that the loops that represent the classical XY model are characterised by the distribution PD(1), as are the loops of the quantum XY model [36]. However, these heuristics also show that the loops representing the classical Heisenberg model are characterised by PD(\(\frac{3}{2}\)) while the loops of the quantum Heisenberg model are PD(2) [23, 36]. Right now, this looks curious.
It is perhaps worth emphasising that the Poisson–Dirichlet distribution is expected to characterise loop soups only in dimensions three and higher. The behaviour in dimension two is also interesting and partially understood, see [4, 8, 18]. There may be a Berezinskii–Kosterlitz–Thouless phase where loop correlations have powerlaw decay instead of exponential. A separate topic is the critical behaviour of twodimensional loop soups, that is characterised by conformal invariance and SchrammLöwner evolution; there have been many impressive results in recent years, but we do not discuss this here.
The article is organised as follows. The notation is summarised in Sect. 2 for the comfort of the reader. The random wire model is introduced in Sect. 3 and basic properties are established. The Poisson–Dirichlet conjecture is explained in Sect. 4. Our main results, Theorems 5.1 and 5.2, are stated in Sect. 5. The first claim deals with the density of points in long loops, and the second claim is about even loop correlations being given by Poisson–Dirichlet. Section 6 discusses classical spin systems and their relations to the random wire model. We gather the necessary properties in Sect. 7 by summarising and completing the results of [21, 32], and we prove Theorems 5.1 and 5.2.
2 Notation

\({\mathcal G}= ({\mathcal V},{\mathcal E})\) the graph; \({\mathcal V}\) is the set of vertices and \({\mathcal E}\) is the set edges. \({\mathcal G}^{\mathrm{b}} = ({\mathcal V}\cup {\bar{{\mathcal V}}}, {\mathcal E}\cup {\bar{{\mathcal E}}})\) denotes the graph with a boundary; \({\bar{{\mathcal V}}}\) is the set of boundary sites and \({\bar{{\mathcal E}}}\) are edges between \({\mathcal V}\) and \({\bar{{\mathcal V}}}\).

\({\mathcal G}_L = (\Lambda _L,{\mathcal E}_L)\) with \(\Lambda _L = \{L,\dots ,L\}^d \subset {\mathbb Z}^d\) and \({\mathcal E}_L\) the set of nearestneighbours. \({\mathcal G}_L^{\mathrm{b}}\) is the graph with boundary \(\partial \Lambda _L\), given by sites of \({\mathbb Z}^d\) at distance 1 from \(\Lambda _L\).

\({\mathcal W}_{\mathcal G}= \{ {{\varvec{w}}}= ({{\varvec{m}}},{\varvec{\pi }}) \}\) is the set of wire configurations on \({\mathcal G}\), that consists of a link configuration \({{\varvec{m}}}\in {\mathcal M}_{\mathcal G}\subset {\mathbb N}_0^{\mathcal E}\) (with an even number of links touching each site) and a pairing configuration \({\varvec{\pi }}\in {\mathcal P}_{\mathcal G}({{\varvec{m}}})\).

\(n_x({{\varvec{m}}})\) is the local occupancy (or “local time”); it is equal to the number of times that loops pass by the site \(x \in {\mathcal V}\).

\(\lambda ({{\varvec{w}}})\) is the number of loops in the wire configuration \({{\varvec{w}}}\).

\(\alpha \) is the positive “loop parameter”.

\({{\varvec{J}}}= (J_e)_{e \in {\mathcal E}}\) are “edge constants”, or “coupling parameters”.

\(U : {\mathbb N}_0 \rightarrow {\mathbb R}\) is a potential function; \(U(n_x)\) gives the energy of the \(n_x\) wires that cross the site \(x\in {\mathcal V}\).

\({\mathbb P}_{\mathcal G}^{\alpha ,{{\varvec{J}}}}, {\mathbb E}_{\mathcal G}^{\alpha ,{{\varvec{J}}}}\) denote the probability and expectation with respect to wire configurations.

\(Z_{\mathcal G}(\alpha ,{{\varvec{J}}})\) is the partition function and \(p_{\mathcal G}(\alpha ,{{\varvec{J}}})\) is the pressure.

\({\tilde{n}}_x\) is the number of pairs at the site x that belong to long or open loops.

\(E_X({{\varvec{x}}},{{\varvec{q}}})\) is the set of configurations \({{\varvec{w}}}\) where \((x_i,q_i)\) and \((x_j,q_j)\) belong to the same loop iff i, j belong to the same partition element of X.

\({\mathcal X}_{2k}^{\mathrm{even}}\) is the family of set partitions of \(\{1,\dots ,2k\}\) whose elements have even cardinality.

\(M_\theta (X)\) is the probability that k random points on [0, 1] and a random partition chosen with Poisson–Dirichlet distribution PD(\(\theta \)), yield the set partition X.

\(M_\theta ^{\mathrm{even}}(2k) = \sum _{X \in {\mathcal X}_{2k}^{\mathrm{even}}} M_\theta (X)\) is the probability that 2k random points on [0, 1], and a random partition from PD(\(\theta \)), yield an even set partition.
3 Setting
3.1 Links, pairings, wires, and loops
We now define the loops of a wire configuration. This notion is intuitive and it is illustrated in Fig. 2b, even though the proper definition is a bit cumbersome. We consider the set of finite sequences of labeled links \(\bigl ( (e_1,p_1), \dots , (e_\ell ,p_\ell ) \bigr )\) where \(e_i \in {\mathcal E}\) and \(p_i \in \{1,\dots ,m_{e_i}\}\), and such that \(e_i \cap e_{i+1} \ne \emptyset \), \(i = 1,\dots ,\ell \). We identify sequences that are related by cyclicity and inversion; that is, we identify \(\bigl ( (e_2,p_2),\dots ,(e_\ell ,p_\ell ), (e_1,p_1) \bigr )\) and \(\bigl ( (e_\ell ,p_\ell ),\dots ,(e_1,p_1) \bigr )\) with \(\bigl ( (e_1,p_1),\dots ,(e_\ell ,p_\ell ) \bigr )\). After identification, these sequences form a loop of length \(\ell \). In order to define the set of loops of a given wire configuration \({{\varvec{w}}}\), we can start at any link \((e_1,p_1)\); we choose an endpoint x and get the next link as the one that is paired by the pairing \(\pi _x\); we continue until we get back to \((e_1,p_1)\). For the next loop we choose a link that has not been selected yet, and we proceed alike until all links have been exhausted.
3.2 The model of random wires
The interaction potential typically becomes infinite as the local occupancy diverges. It is natural to consider models where the partition function is finite for all choices of \(\alpha \) and \({{\varvec{J}}}\). The first claim of the next proposition gives a sufficient condition.
Proposition 3.1
 (a)The partition function is bounded by$$\begin{aligned} Z_{\mathcal G}(\alpha , {{\varvec{J}}}) \le \exp \biggl \{ {\bar{\alpha }} C \sum _{e\in {\mathcal E}} J_e \biggr \}. \end{aligned}$$
 (b)Let \(\ell ^{\mathrm{max}}_{x_0}({{\varvec{w}}})\) be the length of the longest loop that passes through the site \(x_0\). For all \(n\in {\mathbb N}\) and all \(\eta \ge 0\), we have$$\begin{aligned} {\mathbb P}_{\mathcal G}^{\alpha ,{{\varvec{J}}}}(\ell ^{\max }_{x_0} \ge n) \le \,\mathrm{e}^{\eta n}\, \sum _{k\ge 0} \sum _{\begin{array}{c} x_1,\dots ,x_k \in {\mathcal V} \\ \{x_{i1},x_i\} \in {\mathcal E}\text { for } i = 1,\dots ,k1 \end{array}} \prod _{i=1}^k \bigl ( \,\mathrm{e}^{\,\mathrm{e}^{\eta }\,{\bar{\alpha }} C J_{\{x_{i1},x_i\}}}\,  1 \bigr )^{1/2}. \end{aligned}$$
The upper bound in (b) involves a sum over walks of arbitrary lengths that start at \(x_0\). In many situations, such as graphs with bounded degrees and \(J_e\) bounded uniformly, this sum is convergent when \({{\varvec{J}}}\) is small. Then all loops passing by the site \(x_0\) are small, that is, their lengths are finite uniformly in the size of the graph.
Proof
The main advantage of open boundary conditions is to allow us to introduce the event where a site belongs to long loops, namely that it is connected to the boundary. This is discussed in Sect. 4.
4 Loop Correlations and Poisson–Dirichlet Distribution
We now fix the graph to be a large box in \({\mathbb Z}^d\) with edges given by nearestneighbours. We write \(\Lambda _L = \{L,\dots ,L\}^d\) for the set of vertices, \({\mathcal E}_L\) for the set of nearestneighbours, and \({\mathcal G}_L = (\Lambda _L,{\mathcal E}_L)\) for this graph. We also assume that \(J_e \equiv J\) is constant. In dimensions \(d\ge 3\), and if J is large enough, we expect that macroscopic loops are present and that they are described by a Poisson–Dirichlet distribution.
4.1 Joint distribution of the lengths of macroscopic loops
 For every \({\varepsilon }>0\), we have$$\begin{aligned} \lim _{n\rightarrow \infty } \lim _{L\rightarrow \infty } {\mathbb P}_{{\mathcal G}_L}^{\alpha ,J} \Bigl ( \sum _{j=1}^n \frac{\ell _j({{\varvec{w}}})}{V({{\varvec{w}}})} \in [m{\varepsilon },m+{\varepsilon }] \Bigr ) = 1. \end{aligned}$$(4.3)

For every \(n\in {\mathbb N}\) and as \(L \rightarrow \infty \), the distribution of the vector \(\bigl ( \frac{\ell _1({{\varvec{w}}})}{mV({{\varvec{w}}})}, \dots , \frac{\ell _n({{\varvec{w}}})}{mV({{\varvec{w}}})} \bigr )\) converges to the Poisson–Dirichlet distribution PD(\(\frac{\alpha }{2}\)) restricted to the first n elements.
The heuristics for this conjecture is explained in the next subsection; it also contains the calculation of the Poisson–Dirichlet parameter, \(\theta = \frac{\alpha }{2}\).
4.2 Heuristics and calculation of the Poisson–Dirichlet parameter
In words, we choose a site uniformly at random, then pick two endpoints uniformly at random, and accept the rewiring with probability \(C \alpha ^{1/2}, C, C \alpha ^{1/2}\) according to whether the number of loops increases by 1, stays constant, or decreases by 1. It is clear that \(T_{{\varvec{m}}}\) satisfies the detailed balance condition, and also that the process is irreducible on \({\mathcal P}_{\Lambda _L}({{\varvec{m}}})\) for a fixed \({{\varvec{m}}}\).
Next, we look at the resulting process on partitions. We get a splitmerge process with a priori complicated rates. But we can discard all rewirings that involve microscopic loops, as they have negligible effect in the infinitevolume limit. Much more interesting are changes that affect macroscopic loops. If we select endpoints belonging to different loops, the rewiring always merges them. If we select endpoints belonging to the same loop, the rewiring may split it, or just rearrange it (this is analogous to \(0 \leftrightarrow 8\)). The essence of the conjecture is that macroscopic loops merge well, and the number of pairs of endpoints that allow two macroscopic loops \(\gamma ,\gamma '\) to merge is approximately equal to \(c \ell _\gamma \ell _{\gamma '}\) for a constant c that is independent of \(\gamma ,\gamma '\). Further, the number of pairs of endpoints that allow a macroscopic loop \(\gamma \) to split is approximately equal to \(\frac{1}{4} c \ell _\gamma ^2\), with the same constant c as before. The factor \(\frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2}\) is there because pairs within a loop should be counted once, and only half the pairs cause a split and not a rearrangement.
The conclusion of this heuristic is that, as the volume becomes large, the effective splitmerge process on partitions behaves like the standard, meanfield process where two partition elements \(\eta ,\eta '\) merge at rate \(\frac{2c}{\sqrt{\alpha }} \eta \eta '\) and an element \(\eta \) splits at rate \(\frac{c\sqrt{\alpha }}{2} \eta ^2\); moreover, the element is split uniformly. It is known that the Poisson–Dirichlet distribution with parameter \(\theta = \frac{\alpha }{2}\) is the invariant measure for this process [7, 35, 37] (partial results about uniqueness can be found in [16]).
This long heuristics was needed in order to identify the correct parameter. This justifies the above conjecture.
4.3 Poisson–Dirichlet correlations
As we argue below in Sect. 4.4, the probability that points belong to the same loop, knowing that they belong to long loops, is given by the probability that random points in the interval [0, 1] belong to the same partition element with Poisson–Dirichlet distribution. We collect now the relevant formulæ.
Proposition 4.1
Proof
4.4 Loop correlations—Conjectures
We now formulate the Poisson–Dirichlet conjecture in terms of loop correlations. This is more natural in the context of statistical mechanics, and this is the form that we can prove in a special case.
5 Main Results—Long Loops and Their Joint Distribution
Theorem 5.1
 (a)
\(\tilde{m}(d,J)\) is nondecreasing with respect to d and J.
 (b)
\(\tilde{m}(d,J) = 0\) when \(J < 2^{3/2} \log ( 1 + \frac{1}{(2d)^2})\), for arbitrary dimension d.
 (c)
\(\tilde{m}(d,J) = 0\) when \(d=1,2\), for all \(J\ge 0\).
 (d)
For \(d\ge 3\), there exists \(J_{\mathrm{c}}(d)<\infty \) such that \(\tilde{m}(d,J) > 0\) if \(J > J_{\mathrm{c}}(d)\) and \(\tilde{m}(d,J) = 0\) if \(J < J_{\mathrm{c}}(d)\).
The proof of this theorem can be found in Sect. 7.
Theorem 5.1 establishes that a positive fraction of sites are crossed by long loops when \(d\ge 3\) and J is large enough. It is remarkable that \(\tilde{m}(d,J)\) can be proved to be monotone nondecreasing in d and in J. This property is expected to hold for fairly general random wire models; but the present proof, relying as it does on the equivalent XY spin model and its correlation inequalities, cannot be extended easily.
The claim (b) follows from Proposition 3.1 and it holds for more general \(\alpha \) and U. When \(\alpha = 3,4,5,\dots \), and U(n) is defined by Eq. (6.6) with \(N=\alpha \), the claim (c) also holds (its proof uses the continuous symmetry of the corresponding spin system).
Recall the notion of splashing sequences of sites in (4.18). Our second main result is the weaker form of the Poisson–Dirichlet conjecture, see Eq. (4.24).
Theorem 5.2
The proof of this theorem uses the connections to the classical XY model; it can be found in Sect. 7.
Theorem 5.2 gives a lot of information on the structure of long loops: They are present when \(\tilde{m}(d,J)>0\); arbitrary sites have positive probability to belong to them; multiple long loops occur with positive probability. An important aspect of Theorem 5.2 is that it holds for all k with the same constant \(\tilde{m}(d,J)\). This is compatible with the Poisson–Dirichlet distribution PD(\(\theta \)) with \(\theta =1\); this is incompatible with PD(\(\theta \)) with \(\theta \ne 1\) and with most other distributions on partitions. Theorem 5.2 is then a good step forward towards proving that the correlations due to long loops are given by PD(1).
6 Random Wire Representation of Classical O(N) Spin Systems
We show now that the random wire model can be derived as a representation of classical O(N) spin systems. In fact, the case \(N=1\) is close to the random current representation of the Ising model [1, 20, 24]. The general case \(N\in {\mathbb N}\) can be seen as a reformulation of the Brydges–Fröhlich–Spencer loop model [15, 19]; explicit relations between BFS loops and wire configurations can be found in [3].
Proposition 6.1
 (a)If \(N \in {\mathbb N}\), we have$$\begin{aligned} Z_{\mathcal G}^{\mathrm{spin}}({{\varvec{J}}}) = Z_{\mathcal G}(N,{{\varvec{J}}}). \end{aligned}$$
 (b)If \(N=2,3,4,\dots \), we have$$\begin{aligned} \langle \varphi _{x_1}^{(1)} \varphi _{x_1}^{(2)} \dots \varphi _{x_{2k}}^{(1)} \varphi _{x_{2k}}^{(2)} \rangle _{\mathcal G}^{{\varvec{J}}}= \sum _{X \in {\mathcal X}^{\mathrm{even}}_{2k}} \Bigl ( \frac{2}{N} \Bigr )^{X} \frac{1}{2^{2k}} \sum _{{{\varvec{q}}}\in {\mathbb N}^{2k}} {\mathbb E}_{\mathcal G}^{N,{{\varvec{J}}}} \biggl [ 1_{E_X({{\varvec{x}}},{{\varvec{q}}})} \prod _{j=1}^{2k} \frac{1}{n_{x_j}+\frac{N}{2}} \biggr ]. \end{aligned}$$
Recall that \({\mathcal X}^{\mathrm{even}}_{2k}\) is the set of even set partitions of \(\{1,\dots ,2k\}\). It is possible to consider other correlation functions, for instance \(\langle \varphi _x^{(1)} \varphi _y^{(1)} \rangle _{\mathcal G}^{{\varvec{J}}}\). They can be written as ratios of loop partition functions, with the numerator involving “open” configurations of links where \(2n_x\) and \(2n_y\) are odd. See [1, 20] for the Ising random currents and [15, 28] for the related loop model for O(N) spin systems. But these correlations do not have a direct probability meaning and we ignore them in this article.
In the case \(N=1\) we have \(\,\mathrm{e}^{U(n)}\, = 2^n/(2n1)!!\); the denominator is equal to the number of pairings of 2n elements.
Proof
Proposition 6.2
 (a)
\(\displaystyle Z_{{\mathcal G}^{\mathrm{b}}}^{\mathrm{spin},{{\varvec{1}}}}(J) = Z_{{\mathcal G}^{\mathrm{b}}}(N,J)\).
 (b)
If \(N\ge 2\), \(\displaystyle \langle \varphi _x^{(1)} \varphi _x^{(2)} \rangle _{{\mathcal G}^{\mathrm{b}}}^{J,{{\varvec{1}}}} = \tfrac{1}{N} \, {\mathbb E}_{{\mathcal G}^{\mathrm{b}}}^{N,J} \biggl [ \frac{\tilde{n}_x}{n_x+\frac{N}{2}} \biggr ]\).
Proof
7 Correlations of O(2) Spin Systems
We now calculate the correlation function (6.4). The idea is to use Pfister’s theorem on the characterisation of translationinvariant Gibbs states for the O(2) spin model [32]. In this section the graph is \({\mathcal G}_L^{\mathrm{b}}\), that is, a box in \({\mathbb Z}^d\) with boundary conditions.
We can now prove Theorems 5.1 and 5.2.
Proof of Theorem 5.1
For Theorem 5.1 (b), we use Proposition 3.1 (b)—more precisely, we use a straightforward extension to the case of open boundary conditions. We take \({\bar{\alpha }} = \sqrt{2}\) and \(C=2\). The number of random walks of length k and with fixed initial point is equal to \((2d)^k\). This immediately gives the result.
The absence of long loops when \(d=1\) is an easy exercise, and when \(d=2\) it follows from the works of Pfister [31] and Ioffe, Shlosman, and Velenik [26]; see [20, Theorem 9.2] for a clear exposition. Their result is that the infinitevolume Gibbs state is invariant under spin rotations, so \(\tilde{m}(2,J) = \langle \cos (2\phi _0) \rangle _{{\mathbb Z}^2}^{J,0} = 0\). This proves (c).
Notice that the extremal state decomposition (7.5) is only proved for almost all J; but using the claim (a) about monotonicity in J, we get the existence of \(J_{\mathrm{c}}\) as stated in (d). \(\square \)
Proof of Theorem 5.2
By Proposition 6.1 (b), the left side of the equation of Theorem 5.2 is equal to the limits \(L\rightarrow \infty \) then \(n\rightarrow \infty \) of the correlation function \(2^{2k} \bigl \langle \varphi _{x_1^{(n)}}^{(1)} \varphi _{x_1^{(n)}}^{(2)} \dots \varphi _{x_{2k}^{(n)}}^{(1)} \varphi _{x_{2k}^{(n)}}^{(2)} \bigr \rangle _{{\mathcal G}_L}^J \).
Footnotes
 1.
We are grateful to Peter Mörters for the suggestion.
Notes
Acknowledgements
We are grateful to Jakob Björnberg, Jürg Fröhlich, Peter Mörters, CharlesÉdouard Pfister, Vedran Sohinger, Akinori Tanaka, and Yvan Velenik, for useful discussions. We thank the referee for valuable comments. CB is supported by the Leverhulme Trust Research Project Grant RPG2017228.
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