# Spectral Gap Critical Exponent for Glauber Dynamics of Hierarchical Spin Models

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## Abstract

We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the 4-dimensional *n*-component \(|\varphi |^4\) model at the critical point and its approach from the high temperature side, and of the 2-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz–Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the \(|\varphi |^4\) model), the scaling limit of these models in equilibrium.

## 1 Introduction and Main Results

### 1.1 Introduction

Spin systems in equilibrium have been studied by a variety of methods which led to a very complete mathematical description of the physical phenomena occurring in the different regimes of the phase diagrams. This includes in particular a good understanding of the critical phenomena in a wide range of models. Much less is known about the Glauber dynamics of spin systems. For sufficiently high temperatures, it is well understood that the dynamics relaxes exponentially fast towards the equilibrium measure. For the Ising model, the much more difficult question of fast relaxation in the entire uniqueness regime was addressed in [22, 46, 50, 51]. In the phase transition regime, at least for scalar spins, the dynamical behaviour is governed by the interface motion and the relaxation becomes much slower. In particular, the relaxation time diverges as the system size increases, but the dynamical scaling depends strongly on the choice of the boundary conditions. We refer to [49] for a review, as well as to [21, 44] for more recent results. In the vicinity of the critical point, strong correlations develop and as a consequence the dynamic evolution slows down but is no longer driven by phase separation. Even though the critical dynamical behaviour has been well investigated in physics [36], mathematical results are scarce. The only cases for which polynomial lower bounds on the relaxation or mixing times are known are the two-dimensional Ising model [45], exactly at the critical point, the Ising model on a tree [27], both without sharp exponent, and the mean-field Ising model which is fully understood [26, 42].

The goal of this paper is to investigate the dynamical relaxation of hierarchical models near and at the critical point by deriving the scaling of the spectral gap in terms of the temperature (or the equivalent parameter of the model) and the system size.

Since their introduction by Dyson [28] and the pioneering work of Bleher–Sinai [11], hierarchical models have been a stepping stone to develop renormalisation group arguments. At equilibrium, sharp results on the critical behaviour of a large class of models have typically been obtained first in a hierarchical framework and then later been extended to the Euclidean lattice. For the equilibrium problem, the hierarchical framework results in a significant technical simplification, but the results and methods have turned out to be surprisingly parallel to the case of the Euclidean lattice \(\mathbb {Z}^d\). This point of view is discussed in detail in [9], to which we also refer for an overview of results and references. Building on the results for the hierarchical set-up for the equilibrium problem, we derive recursive relations on the spectral gap after one renormalisation step. This enables us to obtain sharp asymptotic behaviour of the spectral gap for large size Sine-Gordon model in the rough phase (Kosterlitz–Thouless phase) and for the \(|\varphi |^4\) model in the vicinity of the critical point. The scaling coincides in both cases with the one of the hierarchical free field dynamics (with a logarithmic corrections for the \(|\varphi |^4\) model) which describes the equilibrium scaling limit of these models. Renormalisation procedures have already been used to analyze spectral gaps for Glauber dynamics, see e.g., [49], but the renormalisation scheme used in this paper is different and allows to keep sharp control from one scale to the next.

After recalling the definitions of the hierarchical models and presenting the results of this paper in Sect. 1.4, we implement, in Sect. 2, the induction procedure to control the spectral gap after one renormalisation step. We believe that our method could be extended beyond the hierarchical models, thus the induction is described in a general framework under some assumptions which can then be checked for each microscopic models. This is completed in Sect. 3 for the hierarchical \(|\varphi |^4\) model, and in Sect. 4 for the hierarchical Sine-Gordon and the Discrete Gaussian models. Proving these assumptions requires establishing stronger control on the renormalised Hamiltonians in the *large field region* than needed when studying the renormalisation at equilibrium (convexity instead of probabilistic bounds). Such convexity for large fields is the main challenge to extend the method of this paper beyond hierarchical models.

### 1.2 Spectral gap

*M*be a symmetric matrix of spin couplings acting on \(\mathbb {R}^\Lambda \). We consider possibly vector-valued spin configurations \(\varphi = (\varphi _x^i)_{x\in \Lambda , i=1,\dots , n} \in \mathbb {R}^{n\Lambda } = \{ \varphi : \Lambda \rightarrow \mathbb {R}^n\}\), with action of the form

*V*is

*O*(

*n*)-invariant and that

*M*acts by \((M\varphi )_x^i = (M\varphi ^i)_x\) for \(i=1,\dots , n\) and \(x\in \Lambda \). The associated probability measure \(\mu \) has expectation

*H*is given by the system of stochastic differential equations

*n*-dimensional standard Brownian motions. (The continuous Glauber dynamics is also referred to as overdamped Langevin dynamics; to keep the terminology concise we use the term Glauber dynamics in the continuous as well as in the discrete case.) By construction, the measure \(\mu \) defined in (1.2) is invariant with respect to this dynamics. Its relaxation time scale is controlled by the inverse of the spectral gap of the generator of the Glauber dynamics (see, for example, [2, Proposition 2.1]). By definition, the spectral gap is the largest constant \(\gamma \) such that, for all functions \(F: \mathbb {R}^{n\Lambda } \rightarrow \mathbb {R}\) with bounded derivative,

*M*and

*V*, when the size of the domain \(\Lambda \) diverges. For statistical mechanics, the setting of primary interest is a finite domain of a lattice or a torus \(\Lambda = \Lambda _N \subset \mathbb {Z}^d\) whose size tends to infinity, and a short-range spin coupling matrix

*M*, such as the discrete Laplace operator \(-\Delta \) on \(\Lambda \). The discrete Laplace operator has a nontrivial kernel. This degeneracy must be removed through boundary conditions or an external field (mass term). For example, for a cube of side length

*D*with Dirichlet boundary conditions, the smallest eigenvalue is of order \(D^{-2}\). In the hierarchical set-up that we consider, we impose an external field instead of boundary conditions whose size is such that the smallest eigenvalue is at least of order \(D^{-2}\).

*M*, the spectral gap \(\gamma \) of the generator of the Langevin dynamics is equal to the minimal eigenvalue of

*M*(assuming that it is positive) by explicit diagonalisation of (1.3). More generally, for

*V*any strictly convex potential satisfying \(V''(\varphi ) \geqslant c > 0\) uniformly in \(\varphi \), the Bakry–Emery criterion [3] implies that

*M*. Under these conditions, \(\mu \) actually satisfies a logarithmic Sobolev inequality with the same constant. In particular, under these assumptions, the dynamics relaxes quickly, in time of order 1.

The situation is much more subtle when the potential *V* is non-convex. Indeed, as the potential becomes sufficiently non-convex, the static measure \(\mu \) typically undergoes phase transitions. In fact for unbounded spin systems on a lattice, the relaxation of the Glauber dynamics has been controlled only in the uniqueness regime under some assumptions on the decay of correlations [12, 13, 39, 41, 53] (see also [52] for conservative dynamics). By considering hierarchical models, we are able to show that the spectral gap decays polynomially in the vicinity of a phase transition. The idea is to decompose the measure into renormalised fields such that at each scale, conditioned on a block spin field, the renormalised potential remains strictly convex. By induction, we then obtain a recursion on the spectral gaps of the renormalised measures.

Before stating the results, we first turn to the definition of the hierarchical models.

### 1.3 Hierarchical Laplacian

There is some flexibility in the choice of the hierarchical field; the precise choice is not significant. Let \(\Lambda = \Lambda _N\) be a cube of side length \(L^N\) in \(\mathbb {Z}^d\), \(d \geqslant 1\), for some fixed integer \(L>1\) and *N* eventually chosen large. For scale \(0\leqslant j \leqslant N\), we decompose \(\Lambda \) as the union of disjoint blocks of side lengths \(L^j\) denoted \(B \in {\mathcal {B}}_j\); see Fig. 1. In particular, \({\mathcal {B}}_0 = \Lambda \) and the unique block in \({\mathcal {B}}_N\) is \(\Lambda _N\) itself. The blocks have the structure of a *K*-ary tree with \(K=L^d\), height *N* and the leaves are indexed by the sites \(x \in \Lambda _N\).

*j*and \(x\in \Lambda \), let \(B_{j}(x)\) be the block in \({\mathcal {B}}_j\) containing

*x*. As in [9, Chapter 4], define the block averaging operators, which are the projections

*hierarchical*if it is diagonal with respect to this decomposition. To obtain a hierarchical Green function with the scaling of the Green function of the usual Laplace operator, we choose the hierarchical Laplace operator on \(\Lambda \) to be

*N*, and \(A \asymp B\) denotes that

*A*/

*B*and

*B*/

*A*are bounded by

*N*-independent constants. On the other hand, the hierarchical Laplacian has coarser small distance behaviour than the lattice Laplacian. For a more detailed introduction to the hierarchical Laplacian, as well as discussion of its relation to the lattice Laplacian, see [9, Chapters 3–4].

### 1.4 Models and results

In Sect. 2, we are going to develop a quite general multiscale strategy to estimate the spectral gap of (critical) spin systems by using a renormalisation group approach. We will then apply this method to the *n*-component \(|\varphi |^4\) model and the Sine-Gordon model as well as the degenerate case of the Discrete Gaussian model. These models correspond to choices of the potential *V* defined now. In the setting of the hierarchical spin coupling, we study the critical region of the \(|\varphi |^4\) model and the rough phase of the Sine-Gordon and Discrete Gaussian models. These are both settings for which the renormalisation group method is well developed for the equilibrium case, and we use this as input.

#### 1.4.1 Ginzburg–Landau–Wilson \(|\varphi |^4\) model

*n*-component \(|\varphi |^4\) model is defined by the double-well potential (if \(n=1\)), respectively Mexican hat shaped potential (if \(n\geqslant 2\)),

*O*(

*n*) symmetry. The spatial dimension \(d=4\) is critical for this model (see, e.g., [9]). The following theorem quantifies the decay of the spectral gap in the four-dimensional hierarchical \(|\varphi |^4\) model when approaching the critical point from the high temperature side.

### Theorem 1.1

*n*-component \(|\varphi |^4\) model on \(\Lambda _N\) with dimension \(d=4\) (as defined above). Let \(L \geqslant L_0\), and let \(g>0\) be sufficiently small. There exists \(\nu _c = \nu _c(g,n) = -C(n+2)g + O(g^2)\) and a constant \(\delta \geqslant 1\) (independent of

*n*) such that for \(t_0 \geqslant t \geqslant cL^{-2N}\), where \(t_0\) is a small constant,

*N*is sufficiently large. In particular, \(t\geqslant cL^{-2N}\) is allowed to depend on

*N*.

The proof is postponed to Sect. 3. The same proof also implies easily that for \(t \geqslant t_0\) the gap is of order 1, but since we are interested in the more delicate approach of the critical point, we omit the details. Together with this, Theorem 1.1 implies that for the \(|\varphi |^4\) model, the spectral gap is of order 1 in the high temperature phase, \(\nu > \nu _c\) independently of *N*, and as the critical point is approached the spectral gap scales like that of the free field, with a logarithmic correction. We expect that \(\gamma \sim Ct(-\log t)^{-z}\) for a universal critical exponent \(z = z(n) \geqslant \frac{n+2}{n+8}\), which our method does not determine (see also [36]). The upper bound follows easily from the estimates derived at equilibrium in [9, Theorem 4.2.1] and we also use the renormalisation group flow constructed in [9] as input to prove the lower bound (see also [33]). References for the renormalisation group analysis of the \(|\varphi |^4\) model on \(\mathbb {Z}^4\), with different approaches, include [31, 34, 37, 38] and [5, 6, 7, 8, 17, 18, 19, 20].

#### 1.4.2 Sine-Gordon model

*V*was, e.g., the double well potential \(V(\varphi ) = \varphi ^4-\varphi ^2\) instead of a periodic potential as above, then the corresponding measure has a uniform spectral gap for any \(\beta >0\) sufficiently small (see, e.g., [4]). The following theorem shows that this is not the case for periodic potentials: the spectral gap decreases to 0. Thus that the resulting models are critical, in the sense of slow decay of correlations, is also reflected in their dynamics.

For the statement of the theorem, denote by \({\hat{V}}(q) = (2\pi )^{-1} \int _{-\pi }^\pi e^{iq\varphi } V(\varphi ) \, d\varphi \) the Fourier coefficient of the \(2\pi \)-periodic function *V*, and let \(\sigma = 1-L^{-2}\) be the constant in (1.9) with dimension \(d=2\).

### Theorem 1.2

*N*is sufficiently large.

The Sine-Gordon model is dual to a Coulomb gas model (see, e.g., [16, 32]). Under this duality, the inverse temperature of the Coulomb gas model is proportional to the temperature \(1/\beta \) of the Sine-Gordon model. We here primarily view the Sine-Gordon model as a spin model, rather than as a description of the Coulomb gas, and therefore choose \(\beta \) instead of \(1/\beta \) in (1.12). Note that the usual normalisation of the logarithm in (1.9) is \(c_N - \frac{1}{2\pi } \log |x| + O(1)\) for the Laplace operator on \(\mathbb {Z}^2\). For this normalisation of the hierarchical Laplace operator, the hierarchical critical inverse temperature becomes \(1/\beta = 8\pi \). This is only approximately true in the Euclidean model because of a field-strength (stiffness) renormalisation which is not present in the hierarchical model. For the critical inverse temperature \(\beta = \sigma /(4\log L)\), we expect that \(\gamma \sim C L^{-2N} N^{-z}\) for a universal critical exponent \(z>0\). For the presence of logarithmic corrections to the free field scaling in the static case, see [30]. Our theorem uses the set-up for the renormalisation group for this model of [16] (see also [48]). References for the Sine-Gordon model on \(\mathbb {Z}^2\) include [32] and [23, 24, 25, 29, 30, 47].

#### 1.4.3 Discrete Gaussian model

### Theorem 1.3

*N*is sufficiently large.

## 2 Induction on Renormalised Brascamp–Lieb Inequalities

*X*with inner product \((\cdot ,\cdot )\) satisfies a Brascamp–Lieb inequality with quadratic form \(D :X \rightarrow X\) if for all smooth functions

*F*,

### 2.1 Hierarchical decomposition

*O*(

*n*)-invariant vector-valued case, in which all operators act separately on each component, and we use the same notation in this case. Thus the Laplacian and the covariances act on the space \(X_0 = \mathbb {R}^{n\Lambda }\).

*F*be such a function of class \(C^2\) written as

*F*can be extended as a smooth function on the whole of \(\mathbb {R}^{n\Lambda }\) by setting, for example,

*F*, we will consider the gradient and the Hessian of

*F*only in the directions spanned by Open image in new window so that we set

*F*has been extended in \(\mathbb {R}^{n\Lambda }\).

### 2.2 Renormalised measure

### 2.3 One step of renormalisation

For the remainder of the section, we fix a scale \(j \in \{0,1,\dots , N\}\), and consider a single renormalisation group step from scale *j* to scale \(j+1\) when \(j<N\), and a final estimate when \(j=N\). To simplify the notation, we usually omit the scale index *j* and write \(+\) in place of \(j+1\). In particular, we write \(C=C_j\), \(V=V_j\), \(\mu = \mu _{j}\), \(\mu _{+} = \mu _{j+1}\), and so on. Let \(X = X_j \subseteq X_0\) be the image of *C* and denote by *Q* the orthogonal projection from \(X_0\) onto *X*. We need the following assumptions.

For \(j<N\), in the assumptions below, \(D_+=D_{j+1}\) is the matrix associated with a quadratic form for a Brascamp–Lieb inequality for the measure \(\mu _+\) [see (2.19)], and we set \(D_{N+1}=0\). Throughout the paper, inequalities between operators and matrices are interpreted in the sense of quadratic forms.

**A1. Non-convexity of potential**There is a constant \(\varepsilon = \varepsilon _j < 1\) such that uniformly in \(\varphi \in X\),

**A2. Coupling of scales**The images of

*C*and \(C_+\) contain all directions on which \(D_+\) is nontrivial, more precisely

**A3. Symmetry**For all \(\varphi \in X\),

The most significant assumption is (2.16), which will be seen to ensure that the fluctuation field measure given the block spin field is uniformly strictly convex. The more technical assumptions (2.17) and (2.18) are very convenient (and obvious in the hierarchical setting (2.3)) but seem less fundamental. We use (2.16) in Lemma 2.7 and (2.60), (2.17) in (2.56), and (2.18) in (2.59).

Under the above assumptions, we relate the Brascamp–Lieb inequality for \(\mu _+\) to that for \(\mu \).

### Theorem 2.1

Iterating this theorem starting from \(j=N\) gives the Brascamp–Lieb inequality for the original measure \(\mu _{0}\) as follows. In particular, the spectral gap of \(\mu _0\) is bounded by the inverse of the largest eigenvalue of the matrix \(D_0\).

### Corollary 2.2

### Proof

*j*replaced by \(j+1\). This means that (2.19) holds for

*j*and Assumptions (A1)–(A3) also hold by assumption of the corollary. Theorem 2.1 and the inductive assumption imply that \(\mu _j\) satisfies the Brascamp–Lieb inequality with

*j*. \(\square \)

### Corollary 2.3

Under the assumptions of the previous corollary, the measure \(\mu _0\) satisfies a spectral gap inequality with inverse spectral gap less than the largest eigenvalue of the matrix \(D_0\).

### Proof

In Sects. 3 and 4, Assumptions (A1)–(A3) will be checked for the different hierarchical models in order to derive the scaling of the spectral gap from the previous corollary.

### Remark 2.4

### 2.4 Proof of Theorem 2.1

*j*as \(\zeta +\varphi \) where \(\varphi \in X_+\) is the

*block spin field*at the next scale \(j+1\) and \(\zeta \in X\) is the

*fluctuation field*at scale

*j*. More precisely, recall that

*C*.

### Lemma 2.5

*F*with gradient in \(L^2(\mu )\), one has

### Lemma 2.6

*F*with gradient in \(L^2(\mu )\), one has

### Proof of Theorem 2.1

For \(j<N\), the proof is immediate by combining the decomposition (2.31) and the previous two lemmas. For \(j=N\), the claim follows directly from Lemma 2.5 only. \(\square \)

#### 2.4.1 Proof of Lemma 2.5

*X*, the image of

*C*[see (2.6) in the hierarchical case]. As a subspace of the Euclidean vector space \(X_0\), the space

*X*has an induced inner product which we also denote by \((\cdot ,\cdot )\), and an induced surface measure, which is equivalent to the Lebesgue measure of the dimension of

*X*. The measure \(\mu _\varphi \) has density proportional to \(e^{-H_\varphi (\zeta )}\) with respect to this measure given by

*X*we can regard

*C*as an invertible symmetric operator \(X \rightarrow X\).) For a function \(F: X_0 \rightarrow \mathbb {R}\) and \(\varphi \in X_0\), the function \(F_\varphi : X \rightarrow \mathbb {R}\) is defined by \(F_\varphi (\zeta ) = F(\varphi +\zeta )\).

### Lemma 2.7

### Proof

*X*, the Hamiltonian \(H_\varphi \) associated with \(\mu _\varphi \) is strictly convex on

*X*, with

*C*is invertible on

*X*and that \(QC = CQ= C\). The Brascamp–Lieb inequality (A.4) implies the inequality. \(\square \)

### Proof of Lemma 2.5

*Q*as the orthogonal projection onto the image of

*C*so that \(\nabla _X\) can be replaced by \(\nabla \). \(\square \)

#### 2.4.2 Proof of Lemma 2.6

We first state a technical lemma.

### Lemma 2.8

### Proof

### Lemma 2.9

Applying the expectation \(\mathbb {E}_{\mu _+}( \cdot )\) on both sides and substituting the result into (2.40), this completes Lemma 2.6.

### Proof of Lemma 2.9

*X*. We define \(L_\varphi \) to be the self-adjoint generator of the Glauber dynamics for the conditional measure \(\mu _\varphi \) on

*X*, i.e.,

*C*and with \({\mathcal {L}}_\varphi C\) by (2.18), the operator \(M_\varphi \) acts on \(L^2(\mu _\varphi ) \otimes X\) and is self-adjoint. From (2.54) and the Cauchy-Schwarz inequality, we finally obtain

*C*are positive operators, using Assumption (2.16), it follows that as operators on \(L^2(\mu _\varphi ) \otimes X\),

## 3 Hierarchical \(|\varphi |^4\) Model

In this section, we apply Corollaries 2.2 and 2.3 to the hierarchical \(|\varphi |^4\) model. Throughout this section, the dimension is fixed to be \(d=4\). Nevertheless, we sometimes write *d* to emphasise that a factor 4 arises from the dimension \(d=4\) rather than from the exponent of \(|\varphi |^4\).

### 3.1 Renormalisation group flow

*g*and \(\nu \)), we decompose

*B*. Explicitly, for a block \(B \in {\mathcal {B}}\), denote by \(i_B: \mathbb {R}^n \rightarrow \mathbb {R}^{nB}\) the linear map that sends \(\varphi \in \mathbb {R}^n\) to the constant field \(\varphi : B \rightarrow \mathbb {R}^n\) with \(\varphi _x = \varphi \) at every \(x \in B\). Then \(V_j(B) \circ i_B\) is a function of a single variable in \(\mathbb {R}^n\) induced by \(V_j(B)\). In particular using (2.10) one can view \(V_j(B)\) as a function in \(\mathbb {R}^{nB}\), so that for any \({{\dot{\varphi }}} \in X_j(B)\) taking the constant value \({{\dot{\varphi }}}_B \in \mathbb {R}^n\),

*j*-dependent) constant and \(j_m = \lfloor {\log _L m^{-1}} \rfloor \) is the mass scale. We stress the fact that if the field is constant on

*B*then

*B*| [see also (3.5)].

*fluctuation field scale*\(\ell _j\) and the

*large field scale*\(h_j\) by

*B*; (ii) the function \(F(B) \circ i_B\) is the same for any block

*B*; and (iii) the function

*F*(

*B*) is invariant under rotations, i.e., \(F(\varphi ,B) = F(T\varphi ,B)\) for any \(T \in O(n)\) acting on \(\varphi \in \mathbb {R}^{n\Lambda }\) by \((T\varphi )_x = T\varphi _x\); see [9, Definition 5.1.5].

### Theorem 3.1

- 1.The full renormalised potential \(V_j\) defined by (3.2) satisfies: for all \(\varphi \) that are constant on
*B*,$$\begin{aligned} e^{-V_j(B,\varphi )} = e^{-u_j|B|}(e^{-{\hat{V}}_j(B,\varphi )}(1+{\hat{W}}_j(B,\varphi )) + {\hat{K}}_j(B,\varphi )). \end{aligned}$$(3.11) - 2.The sequence \((g_j,\nu _j)\) of coupling constants satisfies \((g_0,\nu _0)=(g,\nu -m^2)\), andwhere \(\beta _j = \beta _0^0(1+m^2L^{2j})^{-2}\) for an absolute constant \(\beta _0^0>0\) and \(j_m = \lfloor {\log _Lm^{-1}} \rfloor \).$$\begin{aligned} g_{j+1} = g_j - \beta _j g_j^2 + O(2^{-(j-j_m)_+}g_j^3), \qquad 0 \geqslant L^{2j}\nu _j = O(2^{-(j-j_m)_+}g_j), \end{aligned}$$(3.12)
- 3.The functions \({\hat{K}}_j\) satisfy \({\hat{K}}_0=0\) and$$\begin{aligned} \sup _{\varphi \in \mathbb {R}^n} \max _{0\leqslant \alpha \leqslant 3} h_j^{\alpha } |\nabla ^\alpha ({\hat{K}}_j(B) \circ i_B)(\varphi )|&= O(2^{-(j-j_m)_+}g_j^{3/4}), \end{aligned}$$(3.13)where \(\ell _j = L^{-j}\) and \(h_j = L^{-j} g_j^{-1/4}\).$$\begin{aligned} \max _{0\leqslant \alpha \leqslant 3} \ell _j^{\alpha } |\nabla ^\alpha ({\hat{K}}_j(B) \circ i_B)(0)|&= O(2^{-(j-j_m)_+}g_j^{3}), \end{aligned}$$(3.14)
- 4.The relation between \(t = \nu - \nu _c(g) >0\) and \(m^2>0\) satisfies, as \(t \downarrow 0\),$$\begin{aligned} m^2 \sim C_g t(\log t^{-1})^{-(n+2)/(n+8)}. \end{aligned}$$(3.15)

In the above theorem and everywhere else, the error terms \(O(\cdot )\) are uniform in the scale *j*. The theorem is mainly proved and explained in [9]. For our application to the analysis of the spectral gap of the Glauber dynamics, it is however more convenient to use a slightly different organisation than that used in [9]. It is here better to use the decomposition (2.3) instead of (2.2) (used in [9]). We translate between the conventions in [9] and those used in the statement of Theorem 3.1 in Appendix B and also give precise references there.

A variant of the theorem implies the following asymptotic behaviour of the susceptibility as the critical point is approached.

### Corollary 3.2

*o*(1) tending to 0 as \(L^{2N}m^2 \rightarrow \infty \), and \({{\,\mathrm{Var}\,}}_\mu \) denotes the variance under the full \(|\varphi |^4\) measure as in (1.2).

*F*as defined in the corollary,

*F*by definition of the spectral gap.

### 3.2 Small field region

### Corollary 3.3

### 3.3 Large field region

Using the small field estimates as input, we are going to prove the following estimate for the large field region.

### Theorem 3.4

To prove Theorem 1.1, we will only use the conclusion \(\varepsilon _j \geqslant 0\) from Theorem 3.4. However, in order to prove Theorem 3.4, it is convenient that the \(\varepsilon _j\) do not become too small. The elementary proof of the following estimate is given in Appendix B.

### Lemma 3.5

The sequence \((\varepsilon _j)\) defined in Theorem 3.4 satisfies \(\varepsilon _j \geqslant c g_j\) for all \(j\in \mathbb {N}\).

*j*. For \(j=0\), the estimate (3.28) can be checked directly from (3.23) and \(\nu \geqslant \nu _c(g) = -O(g)\), which imply that

### Lemma 3.6

### Proof

The following proposition now advances the induction and thus proves Theorem 3.4.

### Proposition 3.7

*j*will be fixed we usually drop the

*j*and write \(+\) instead of \(j+1\). To set-up notation, we fix a block \(B_+ \in {\mathcal {B}}_{+}\) and write \(V(B_+) = \sum _{B \in {\mathcal {B}}_j(B_+)} V(B)\). By the hierarchical structure, \({{\,\mathrm{Hess}\,}}V(B_+)\) is a block diagonal matrix indexed by the blocks \(B \in {\mathcal {B}}(B_+)\), and we will always restrict the domain to \(X_j(B_+)\), the space of fields constant inside the small blocks

*B*. On this domain, \(V(B_+)\) can be identified with a function of \(L^d\) vector-valued variables while \(V_+(B_+)\) has domain \(X_{+}(B_+)\) and can be identified with a function of a single vector-valued variable. The covariance operator

*C*and the projection

*Q*operate naturally on \(X(B_+)=X_j(B_+)\) and can be identified with diagonal matrices indexed by blocks \(B \in {\mathcal {B}}(B_+)\); in particular, they are invertible on \(X(B_+)\). By the definition of \(V_+\) in (3.2), together with the hierarchical structure of

*C*, it follows that

*C*denotes the restriction of

*C*to \(X(B_+)\))

*j*blocks in \(B_+\).

*C*acts diagonally.

Further recall that \(\varphi \) is constant on \(B_+\). By symmetry and uniqueness of the minimiser, we see that \(\zeta ^0\) has to be constant not only in each small block *B*, but in each \(B_+\), i.e., \(\zeta ^0 \in X_{+}(B_+)\). In the following lemma, the block \(B_+\) is fixed and \(\varphi \) and \(\zeta ^0\) are both in \(X_{+}(B_+)\) so that we may identify them with variables in \(\mathbb {R}^n\).

### Lemma 3.8

Let \(|\varphi |\geqslant h_+\). Then \(|\varphi +\zeta ^0| \geqslant h_+(1 - O(g^{1/2}))\).

### Proof

*C*acts by multiplying each of these blocks by the same constant \(O(\vartheta ^2 L^{2j})\). Hence \(C\nabla V(B_+,\varphi ')\) is a block vector with all blocks equal to \(O(\vartheta ^2 L^{2j})\nabla V(B,\varphi ')\) where

*B*is any of the block in \({\mathcal {B}}(B_+)\). We denote by \(|C\nabla V(B_+,\varphi ')|_\infty \) the value in any of these blocks. Now (3.27) implies that, for \(\varphi '\) constant on \(B_+\) with \(|\varphi '|\leqslant h_+\),

In the following lemma, \(\zeta \in X(B_+)\) is the fluctuation field under the measure with expectation \(\langle \cdot \rangle _{H_\varphi }\). Thus \(\zeta \) is constant in any small block *B*, but unlike the minimiser \(\zeta ^0\) the field \(\zeta \) is not constant in \(B_+\).

### Lemma 3.9

### Proof

*H*has the same Hessian as \(H_\varphi \). From the information that the minimiser of

*H*is 0, we obtain a bound on the random variable \(\zeta \) as follows. Using that \({{\,\mathrm{Hess}\,}}H \geqslant \frac{1}{2} C^{-1}\) as quadratic forms and that \(C_{xx} \leqslant \vartheta ^2\ell ^2\) for all \(x\in \Lambda \) by definition, the Brascamp–Lieb inequality (A.5) for the measure \(\langle \cdot \rangle _H\) with density proportional to \(e^{-H}\) implies

Next we use the following estimate on \({{\,\mathrm{Hess}\,}}V_+(B_+)\).

### Lemma 3.10

Note that \({{\,\mathrm{Hess}\,}}V(B_+,\varphi +\zeta )\) are both diagonal matrices indexed by \(B \in {\mathcal {B}}_+\), with constant entries on each block *B*. In fact, *C* is proportional to the identity matrix on \(X(B_+)\).

### Proof

*C*are both (block) diagonal matrices, the term inside the expectation can be written as

The next lemma completes the proof of Proposition 3.7.

### Lemma 3.11

### Proof

*B*and taking a union bound over the \(L^d\) blocks \(B \in {\mathcal {B}}(B_+)\) we get that \(\max _x |\zeta _x-\zeta ^0| \geqslant \frac{1}{4} h_+\) with probability at most \(2L^d e^{-c (\vartheta g)^{-1/2}}\). Since \(|\varphi +\zeta ^0| \geqslant h_+(1-O(g^{1/2})) \geqslant \frac{3}{4} h_+\) by Lemma 3.8, together with the assumption \(|\varphi | \geqslant h_+\), we conclude that \(\min _x|\varphi +\zeta _x| \geqslant \frac{1}{2} h_+\) with probability at least \(1-2L^d e^{-c(\vartheta g)^{-1/2}}\). Thus (3.56) holds with at least this probability.

### 3.4 Proof of Theorem 1.1

### Lemma 3.12

The elementary proof requires some notation from [9]; we therefore postpone it to Appendix B.

### Proof of Theorem 1.1

## 4 Hierarchical Sine-Gordon and Discrete Gaussian Models

In this section, we apply Corollaries 2.2 and 2.3 to the hierarchical versions of the Sine-Gordon and the Discrete Gaussian models. This boils down to checking that Assumption (A1) is satisfied along the renormalisation group flow of both models. Throughout this section \(d=2\).

### 4.1 Proof of Theorem 1.2

*B*. The final potential obtained as \(V_{N+1}\) in (2.14) will instead be denoted by \(V_{N,N}\) since it is indexed by the final block \(\Lambda \in {\mathcal {B}}_N\), i.e., \(V_{N,N}(\varphi ) = V_{N,N}(\Lambda _N,\varphi )\), and \(\varphi \) can be seen as an external field. Then each \(V_j(B)\) can be identified with a \(2\pi \)-periodic function on \(\mathbb {R}\) (and analogously for \(V_{N,N}\)). For any such function \(F: S^1 \rightarrow \mathbb {R}\), we use the norm

*F*is \({\hat{F}}(q) = (2\pi )^{-1} \int _0^{2\pi } F(\varphi ) e^{iq\varphi } \, d\varphi \). We write

*B*). Except for the weight

*w*(

*q*), the norm (4.5) is the one used in [16, 48].

### Proposition 4.1

The derivation of this proposition is postponed to Sect. 4.2. We now state consequences of this proposition and prove Theorem 1.2 using these.

### Corollary 4.2

### Proof

### Corollary 4.3

### Proof

### Proof of Theorem 1.2

*j*, and Assumptions (A2) and (A3) always hold. Assumption (A1) follows from Corollary 4.2 which implies that for \(j \leqslant N\)

### 4.2 Proof of Proposition 4.1

The proof of Proposition 4.1 follows as in [16, Chapter 3], with small modifications. Throughout Sect. 4.2, the full covariance matrix \((-\beta \Delta _H+\varepsilon Q_N)^{-1}\) does not play a role and we write \(C = C_j\) for a fixed scale *j*. More generally, we drop the scale index *j* and write \(+\) in place of \(j+1\). We write \(B_+\) for a fixed block in \({\mathcal {B}}_+\) and *B* for the blocks in \({\mathcal {B}}(B_+)\).

### Lemma 4.4

### Proof

### Proof of Proposition 4.1

*B*for the blocks in \({\mathcal {B}}(B_+)\). By definition of the hierarchical model, the Gaussian field \(\zeta \) with covariance \(C=C_j\) is constant in any block \(B \in {\mathcal {B}}_j\) and we thus write \(\zeta _B\) for \(\zeta _x\) with \(x\in B\). We then start from

*X*is a union of blocks \(B\in {\mathcal {B}}(B_+)\). The term with \(|X|=0\) is simply 1. By (4.27) and (4.25), the terms with \(|X|=1\) are bounded by

*B*and the product property of the norm, the terms with \(|X|>1\) give

### 4.3 Proof of Theorem 1.3

We will now reduce the result for the Discrete Gaussian model to that for the Sine-Gordon model. For this, we carry out an initial renormalisation group step by hand, resulting in an effective Sine-Gordon potential for the Discrete Gaussian model. This strategy for the Discrete Gaussian model (and more general models) goes back to [32].

*A*/

*B*is independent of \(\sigma \), and where the Gaussian expectation applies to the field \(\varphi \). We define the effective single-site potential \(V(\psi )\) for \(\psi \in \mathbb {R}\) by

*V*is \(2\pi \)-periodic as in the Sine-Gordon model. This is where the \(2\pi \)-periodicity of the Discrete Gaussian Model is convenient. For \(\psi \in \mathbb {R}\), we also define a probability measure \(\mu _\psi \) on \(2\pi \mathbb {Z}\) by

*V*satisfies the conditions of Theorem 1.2 provided \(\beta \) is sufficiently small, and that the probability measure \(\mu _\psi \) satisfies a spectral gap inequality on \(2\pi \mathbb {Z}\), with constant uniform in \(\psi \). It is clear from the definition (4.38) that

*V*is \(2\pi \)-periodic.

### Lemma 4.5

For \(\beta >0\) small enough, *V* is smooth with \(\Vert V-{\hat{V}}(0)\Vert =O(e^{-1/(2\beta )})\).

### Proof

*V*, we can normalise

*F*such that \({\hat{F}}(0)=1\). Note that subtraction of a constant does not change \(V-{\hat{V}}(0)\). The Fourier coefficients of

*F*are then given by

*C*and the last equality are due to the normalisation \({\hat{F}}(0)=1\). It follows that

### Corollary 4.6

For \(\beta >0\) sufficiently small, the measure \(\mu _r\) has inverse spectral gap \(O(1/\varepsilon )\).

### Proof

The proof is essentially the same as that of Theorem 1.2. The only difference compared to Theorem 1.2 is that we replaced \(C_{\geqslant 0}\) by \(C_{\geqslant 1}\) which does not change the conclusion. For small \(\beta \), the assumption on *V* is satisfied thanks to Lemma 4.5. \(\square \)

The following lemma can be proved, e.g., using the *path method* for spectral gap inequalities; we postpone the elementary proof to Appendix C.

### Lemma 4.7

With the above ingredients, the proof can now be completed as follows.

### Proof of Theorem 1.3

*x*). It follows that

## Notes

### Acknowledgements

We warmly thank Tom Spencer for his contributions to this paper; his input has been crucial. We also thank David Brydges and Gordon Slade for a number of important discussions and for careful reading and many helpful comments on a preliminary version of this paper. Figure 1 is taken from [9]. We acknowledge the support of ANR-15-CE40-0020-01 grant LSD.

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