Log-Sobolev inequality for the continuum sine-Gordon model

TL;DR

This paper develops a multiscale criterion based on the Polchinski equation to establish Log-Sobolev inequalities for the continuum Sine-Gordon model, extending applicability beyond log-concave measures.

Contribution

It introduces a novel multiscale Bakry--Émery criterion leveraging the Polchinski PDE, applicable to non-log-concave measures like the continuum Sine-Gordon model.

Findings

Proves Log-Sobolev inequalities for the massive continuum Sine-Gordon model with 2<6c

Establishes asymptotically optimal inequalities for Glauber and Kawasaki dynamics

Results are independent of recent regularity structures for singular SPDEs

Abstract

We derive a multiscale generalisation of the Bakry--\'Emery criterion for a measure to satisfy a Log-Sobolev inequality. Our criterion relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry--\'Emery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, using our criterion, we prove that the massive continuum Sine-Gordon model with $\beta < 6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory.

Figures

Equations

$${\mathbb{E}}_{{\nu }_0}F\propto {\int }_X{e}^{-\frac{1}{2}(\zeta ,A\zeta )-V_0(\zeta )}F(\zeta )d\zeta .$$

$${\dot{C}}_t=Q_t^2=e^{-tA},C_t={\int }_0^t{\dot{C}}_{s}ds,$$

$$e^{-V_t(\phi )}$$

$${\mathbf{P}}_{s,t}F(\phi )$$

$${\mathbb{E}}_{{\nu }_t}F={\mathbf{P}}_{t,\infty }F(0)$$

$${\mathbb{E}}_{{\nu }_t}g({\mathbf{P}}_{0,t}F)\to g({\mathbb{E}}_{{\nu }_0}F)\text{as t→∞.}$$

$${\mathrm{Ent}}_{{\nu }_0}(F)={\mathbb{E}}_{{\nu }_0}\Phi (F)-\Phi ({\mathbb{E}}_{{\nu }_0}F),\Phi (x)=x\log x,$$

$$Q_t\mathrm{Hess}{V}_t(\phi )Q_t⩾{\dot{\mu }}_t\text{mspace}2.0mu\mathrm{i}\text{mspace}-0.6mu\mathrm{d},\text{where Qt=e−tA/2,}$$

$${\mathrm{Ent}}_{{\nu }_0}(F)⩽\frac{2}{\gamma }{\mathbb{E}}_{{\nu }_0}(\nabla \sqrt{F}{)}^2,\frac{1}{\gamma }={\int }_0^{\infty }{e}^{-\lambda t-2{\mu }_t}dt,$$

$${\partial }_t{V}_t=\frac{1}{2}{\Delta }_{{\dot{C}}_t}{V}_t-\frac{1}{2}(\nabla V_t{)}_{{\dot{C}}_t}^2,$$

$$(u,v{)}_w=\sum _{i,j}{w}_{ij}{u}_i{v}_j,(\nabla F{)}_w^2=(\nabla F,\nabla F{)}_w,{\Delta }_{w}F=(\nabla ,\nabla {)}_{w}F.$$

$${\mathbb{E}}_{{\nu }_0}F\propto {\int }_{{\mathbb{R}}^N}{e}^{-H(\zeta )}F(\zeta )d\zeta ,$$

$$H(\zeta )=\frac{1}{2}(\zeta ,A\zeta )+V_0(\zeta ),\text{with A=λmspace2.0muimspace−0.6mud and V0 convex.}$$

$$\frac{1}{Z}\exp \left[-{\int }_{{\mathbb{R}}^2}\left(\frac{1}{2}\phi (-\Delta \phi )+\frac{1}{2}{m}^2\phi (x{)}^2+2z:\cos (\sqrt{\beta }\phi (x)):\right)dx\right]\prod _{x\in {\mathbb{R}}^2}d\phi (x).$$

$$H_{\varepsilon,L}(\varphi)={\varepsilon^{d}}\sum_{x\in\Omega_{\varepsilon,L}}\biggl{(}{\frac{1}{2}\varphi_{x}(-\Delta^{\varepsilon}\varphi)_{x}+\frac{1}{2}m^{2}\varphi_{x}^{2}+2z_{\varepsilon}\cos(\sqrt{\beta}\varphi_{x})}\biggr{)},$$

$$z_{\epsilon }=z{\epsilon }^{-\beta /4\pi },$$

$${\mathbf{D}}_{\epsilon ,L}(F)=\frac{1}{{\epsilon }^2}\sum _{x\in {\Omega }_{\epsilon ,L}}{\mathbb{E}}_{{\nu }_{\epsilon ,L}}\left[{\left(\frac{\partial F}{\partial {\phi }_x}\right)}^2\right],$$

$$\frac{\partial }{\partial t}{\phi }_x^{\epsilon }=({\Delta }^{\epsilon }{\phi }^{\epsilon }{)}_x+m^2{\phi }_x^{\epsilon }+{\epsilon }^{-\beta /4\pi }2z\sqrt{\beta }\sin (\sqrt{\beta }{\phi }_x^{\epsilon })+\sqrt{2}{\dot{W}}_x^{\epsilon },$$

$${\mathrm{Ent}}_{{\nu }_{\epsilon ,L}}(F)⩽\frac{2}{\gamma (\beta ,z,m,L)}{\mathbf{D}}_{\epsilon ,L}(\sqrt{F}).$$

$$\gamma (\beta ,z,m,L)⩾m^2-O_{\beta }(m^{\beta /4\pi }|z|),$$

$${\mathbf{D}}_{\epsilon ,L}^0(F)=\frac{1}{{\epsilon }^4}\sum _{x\sim y\in {\Omega }_{\epsilon ,L}}{\mathbb{E}}_{{\nu }_{\epsilon ,L}^0}\left[{\left(\frac{\partial F}{\partial {\phi }_x}-\frac{\partial F}{\partial {\phi }_y}\right)}^2\right].$$

$${\mathrm{Ent}}_{{\nu }_{\epsilon ,L}^0}(F)⩽\frac{2}{{\gamma }^0(\beta ,z,m,L)}{\mathbf{D}}_{\epsilon ,L}^0(\sqrt{F}).$$

$${\gamma }^0(\beta ,z,m,L)⩾\frac{(2\pi {)}^2}{L^2}\left(m^2+\frac{(2\pi {)}^2}{L^2}-O_{\beta }(m^{\beta /4\pi }|z|)\right),$$

$${\mathbb{E}}_{{\nu }_0}F\propto {\mathbf{E}}_{C_{\infty }}(e^{-V_0(\zeta )}F(\zeta )),$$

$$e^{-V_t(\phi )}$$

$${\mathbf{P}}_{s,t}F(\phi )$$

$${\mathbb{E}}_{{\nu }_t}F={\mathbf{P}}_{t,\infty }F(0)$$

$${\mathbb{E}}_{{\nu }_t}g({\mathbf{P}}_{0,t}F)\text{is continuous in t∈[0,+∞].}$$

$${\partial }_t{V}_t$$

$$\frac{\partial }{\partial t}{\mathbf{P}}_{s,t}F={\mathbf{L}}_t{\mathbf{P}}_{s,t}F,-\frac{\partial }{\partial s}{\mathbf{P}}_{s,t}F={\mathbf{P}}_{s,t}{\mathbf{L}}_{s}F,(s⩽t),$$

$${\mathbf{L}}_{t}F=\frac{1}{2}{\Delta }_{{\dot{C}}_t}F-(\nabla V_t,\nabla F{)}_{{\dot{C}}_t}.$$

$${\mathbb{E}}_{{\nu }_t}{\mathbf{P}}_{s,t}F={\mathbb{E}}_{{\nu }_s}F(s⩽t),-\frac{\partial }{\partial t}{\mathbb{E}}_{{\nu }_t}F={\mathbb{E}}_{{\nu }_t}{\mathbf{L}}_{t}F.$$

Full text

1 Introduction and results

1.1.  Introduction

Log-Sobolev inequalities are strong inequalities with numerous general consequences, including concentration of measure, relaxation and hypercontractivity of stochastic dynamics, transport inequalities, and others. See [4, 47] for a review. They originate from Quantum Field Theory where Log-Sobolev inequalities were first derived for Gaussian measures as a tool to study non-Gaussian measures in infinite dimensions (Euclidean Quantum Field Theories, EQFTs) [55, 26, 32]. As a consequence of a general new approach, we prove Log-Sobolev inequalities for the massive sine-Gordon model. This is a fundamental example of a non-Gaussian EQFT in two dimensions and its stochastic dynamics is a prototypical example of a singular SPDE.

As Log-Sobolev inequalities provide strong control on the measures they apply to, proving them remains in general a difficult problem even if the equilibrium correlation functions are well understood. This applies especially to strongly correlated measures. For log-concave measures (or measures satisfying a curvature dimension condition), the fundamental Bakry–Émery criterion provides a simple and often quite sharp sufficient condition [3, 2]. In its proof, a Log-Sobolev inequality for a Markov semigroup is derived by integration of local Log-Sobolev inequalities for the same Markov semigroup. Our method also uses local Log-Sobolev inequalities, but for a semigroup that is different from the one for which the Log-Sobolev inequality is proven. Namely our method uses the time-dependent semigroup driven by the Polchinski equation, a version of the renormalisation semigroup. Unlike the original semigroup, this Polchinski semigroup provides a notion of scale and hence we effectively obtain a multiscale version of the Bakry–Émery criterion.

The simplest version of our new Polchinski equation criterion for the Log-Sobolev inequality is stated in Section 1.2. In Example 1.3, we illustrate that it implies the Bakry–Émery criterion. As an application of the new criterion, demonstrating that it remains effective for measures that are far from log-concave, we prove the following theorem for the continuum sine-Gordon model. For a precise statement of this result and related discussion, we refer to Section 1.3. In Section 1.4, we discussed further directions and related results.

Theorem 1.1**.**

The continuum massive sine-Gordon model with $\beta<6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics (under suitable conditions).

Throughout this paper, we make the assumption that all functions considered are Borel measurable and that all functions to which derivatives are applied are continuously differentiable of the required order.

1.2.  Polchinski equation and Log-Sobolev inequality

In this section we state the simplest version of our new criterion for a probability measure to satisfy a Log-Sobolev inequality.

Given a linear space $X\subseteq\mathbb{R}^{N}$ with the induced inner product $(\cdot,\cdot)$, a symmetric matrix $A$ that acts positive definitely on $X$, and a potential $V_{0}:X\to\mathbb{R}$, we consider the probability measure $\nu_{0}$ with expectation

[TABLE]

We call the set $\Lambda=\{1,\dots,N\}$ the index space and the space $X$ the field space; see also Figure 1.1. Let $Q_{t}=e^{-tA/2}$ be the heat semigroup associated with $A$ (acting on elements $\varphi\in X$, i.e., functions $\varphi:\Lambda\to\mathbb{R}$ on the index space), set

[TABLE]

and denote by $\bm{E}_{C_{s}}$ the expectation of the Gaussian measure with covariance $C_{s}$. For $t>s>0$, we define the renormalised potential $V_{t}$, the renormalisation semigroup $\bm{P}_{s,t}$ (acting on functions $F:X\to\mathbb{R}$ on the field space), and the renormalised measure $\nu_{t}$ by

[TABLE]

where $\varphi\in X$, the expectation $\bm{E}_{C_{t}}$ applies to $\zeta$, and it is natural to define $\mathbb{E}_{\nu_{\infty}}F=F(0)$. Essentially equivalently to (1.3), $V_{t}$ solves the Polchinski equation; see (1.10) below.

In what follows, we will impose the following ergodicity assumption on the semigroup $\bm{P}$: For all bounded smooth functions $F:X\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$,

[TABLE]

Like the ergodicity assumption in the Bakry–Émery theory (see [4, 1]), this assumption is qualitative and easily seen to be satisfied in all examples of interest.

The following theorem bounds the Log-Sobolev constant of the measure $\nu_{0}$. For its statement, recall that the relative entropy of $F:X\to\mathbb{R}_{+}$ with respect to $\nu_{0}$ is given by

[TABLE]

where $0\log 0=0$. We write $\nabla$ for the gradient on $X$ and $(\nabla F)^{2}=(\nabla F,\nabla F)$; thus in particular if $X=\mathbb{R}^{N}$ then $(\nabla F)^{2}=\sum_{i=1}^{N}(\frac{\partial F}{\partial\varphi_{i}})^{2}$.

Theorem 1.2**.**

In the set-up above, assume (1.6), let $\lambda>0$ be the smallest eigenvalue of $A$, suppose there are real numbers $\dot{\mu}_{t}$ (possibly negative) such that for all $t\geqslant 0$, as quadratic forms on $X$,

[TABLE]

and define $\mu_{t}=\int_{0}^{t}\dot{\mu}_{s}\,ds$. Then $\nu_{0}$ satisfies the Log-Sobolev inequality

[TABLE]

provided the integral is finite.

The proof of Theorem 1.2, given in Section 2, shares significant elements with the celebrated Bakry–Émery argument, but with the crucial difference that it uses the time-dependent Polchinski semigroup (1.4) rather than the original semigroup, associated with the Dirichlet form $\mathbb{E}_{\nu_{0}}(\nabla F)^{2}$, to decompose the relative entropy. The above version of our criterion relies on the particular decomposition of the matrix $C_{\infty}=A^{-1}$ in terms of the heat semigroup $\dot{C}_{t}=e^{-tA}$. In Section 2, we also consider variations of the criterion that apply to other decompositions.

To apply the theorem, the main task is to verify (1.8). It is not difficult to see that the renormalised potential $V_{t}$ solves the Polchinski equation (see Section 1.4 for its history)

[TABLE]

where we use the notation (and with $w=\mspace{2.0mu}\mathrm{i}\mspace{-0.6mu}\mathrm{d}$ if the argument $w$ is omitted)

[TABLE]

In general, verifying (1.8) is a challenging problem because the Polchinski equation is a non-linear PDE in $N$ dimensions, where in the examples of main interest $N\to\infty$. Nonetheless, we believe that the required estimates are true in many relevant examples, including spin systems near the critical point. In particular, in Section 3, we verify the condition for the continuum sine-Gordon model by analysing the Polchinski equation.

To illustrate our new criterion, we note briefly that (1.8) is not hard to verify for log-concave measures, in which case we recover the Bakry–Émery criterion as a special case.

Example 1.3** (Bakry–Émery criterion).**

Consider a probability measure $\nu_{0}$ with expectation

[TABLE]

where $\operatorname{Hess}H\geqslant\lambda\mspace{2.0mu}\mathrm{i}\mspace{-0.6mu}\mathrm{d}$ holds uniformly for some $\lambda>0$. Equivalently, $\nu_{0}$ can be written as in (1.1):

[TABLE]

It follows that $V_{t}$ is convex for all $t\geqslant 0$ (see, e.g., [10, Theorem 4.3]). Hence $\mu_{t}\geqslant 0$ for all $t$ and thus $\gamma\geqslant\lambda$ in (1.9). This is the Bakry–Émery criterion.

We remark that an alternative proof that $V_{t}$ remains convex for $t>0$ can be deduced from the maximum principle for symmetric tensors [37, Theorem 9.1]. This argument is in fact analogous to the proof that positive Ricci curvature remains positive under the Ricci flow in [37].

Theorem 1.2 can be considered a multiscale version of the Bakry–Émery criterion in which the global convexity assumption $\inf_{\varphi}\operatorname{Hess}V_{0}(\varphi)\geqslant 0$, which is equivalent to $\inf_{t\geqslant 0}\inf_{\varphi}\operatorname{Hess}V_{t}(\varphi)\geqslant 0$, is replaced by the assumption (1.8) on the Hessians of the effective potential $V_{t}$ at each scale $t$. We emphasise that these Hessians are not required be positive definite; and in fact in the example of the continuum sine-Gordon model which we consider in Section 1.3 below, the effective potential remains non-convex at all scales $t>0$. We also emphasise that the application of the heat kernel $Q_{t}$ to $\operatorname{Hess}V_{t}(\varphi)$ in (1.8) has an important smoothing effect. In particular, for the sine-Gordon model, we will see that this smoothing effect is essential when $\beta>4\pi$.

Remark 1.4**.**

We have defined the renormalised potential $V_{t}$ as the convolution solution (1.3) to the Polchinski equation (1.10). Since equivalently $Z_{t}=e^{-V_{t}}$ solves the heat equation $\partial_{t}Z_{t}=\frac{1}{2}\Delta_{\dot{C}_{t}}Z_{t}$, the Polchinski equation has a unique solution under weak conditions. Then one may equivalently solve (1.10) instead of (1.3); for an example for which this is useful, see Section 3.

Remark 1.5**.**

We remark that with the time-dependent metric $g_{t}=e^{+tA}$ on $X$ and $\nabla_{g_{t}}$ and $\Delta_{g_{t}}$ defined as in Riemannian geometry, i.e., $\nabla_{g_{t}}=g_{t}^{-1}\nabla$ and $\Delta_{g_{t}}$ the Laplace-Beltrami operator, one has $\Delta_{\dot{C}_{t}}=\Delta_{g_{t}}$ and $(\nabla F)_{\dot{C}_{t}}^{2}=(\nabla_{g_{t}}F)_{g_{t}}^{2}$. The condition (1.8) then becomes $\operatorname{Hess}_{g_{t}}V_{t}\geqslant\dot{\mu}_{t}g_{t}$.

1.3.  Continuum sine-Gordon model

In Section 3, we apply Theorem 1.2 to prove asymptotically sharp Log-Sobolev inequalities for Glauber and Kawasaki dynamics of the massive continuum sine-Gordon model with $\beta<6\pi$. The massive sine-Gordon model is a fundamental example of a two-dimensional interacting Euclidean Quantum Field Theory, i.e., a non-Gaussian probability measure on $\mathcal{D}^{\prime}(\mathbb{R}^{2})$ sometimes formally written as

[TABLE]

Here $\Delta$ is the Laplacian on $\mathbb{R}^{2}$, and the notation $:$ denotes Wick ordering, i.e., that $z$ is formally multiplied by a divergent constant (making the microscopic potential extremely non-convex); see (1.15)-(1.16) below for the precise definition that we will use. The Glauber dynamics of the sine-Gordon model (also called dynamical sine-Gordon model) can be realised as a singular SPDE that was recently constructed using the theory of regularity structures. References on the sine-Gordon model are provided further below.

For clarity, we consider the model in a lattice approximation of a two-dimensional torus, and prove estimates uniformly in the lattice spacing and in the size of the torus. Therefore, from now on, let $d=2$, let $\Omega_{L}=L\mathbb{T}^{d}$ be the torus of side length $L>0$, and let $\Omega_{\varepsilon,L}=\Omega_{L}\cap\varepsilon\mathbb{Z}^{d}$ be its lattice approximation with mesh size $\varepsilon>0$ (where we always assume $L$ is a multiple of $\varepsilon$). The continuum sine-Gordon model $\nu_{\varepsilon,L}$ in the lattice approximation is the probability measure on $\mathbb{R}^{\Omega_{\varepsilon,L}}$ with density proportional to $e^{-H_{\varepsilon,L}(\varphi)}$ where $H_{\varepsilon,L}$ is defined for $\varphi:\Omega_{\varepsilon,L}\to\mathbb{R}$ by

[TABLE]

with divergent coupling constant

[TABLE]

and where $(\Delta^{\varepsilon}\varphi)_{x}=\varepsilon^{-2}\sum_{y\sim x}(\varphi_{y}-\varphi_{x})$ is the discretised Laplacian, i.e., the sum $y\sim x$ is over nearest neighbour vertices $y$ of $x$ in $\varepsilon\mathbb{Z}^{d}$. Under suitable assumptions, this normalisation ensures that, for $0<\beta<8\pi$, the measures $\nu_{\varepsilon,L}$ converge weakly to a non-Gaussian probability measure on $\mathcal{D}^{\prime}(\mathbb{R}^{2})$ as $\varepsilon\to 0$ and $L\to\infty$; see the discussion after the statement of the theorems below.

Our first theorem is a uniform Log-Sobolev inequality for the Glauber dynamics of the massive sine-Gordon measure $\nu_{\varepsilon,L}$ (with dimension always $d=2$). The Glauber Dirichlet form is given by

[TABLE]

corresponding to the system of SDEs

[TABLE]

where $\dot{W}^{\varepsilon}$ is space-time white noise (with discretised space), i.e., the $(W_{x}^{\varepsilon})_{x\in\Omega_{\varepsilon,L}}$ are independent Brownian motions with quadratic variation $\langle W_{x}^{\varepsilon}\rangle(t)=t/\varepsilon^{2}$.

Theorem 1.6**.**

Fix $\beta<6\pi$, and let $L>0$, $m>0$, and $z\in\mathbb{R}$. Then there is $\gamma(\beta,z,m,L)>0$ independent of $\varepsilon>0$ such that, for all $F\geqslant 0$,

[TABLE]

Moreover, there is $\delta_{\beta}>0$ such that if $Lm\geqslant 1$ and $|z|m^{-2+\beta/4\pi}\leqslant\delta_{\beta}$, then

[TABLE]

where the constant $O_{\beta}$ depends on $\beta$ only (and is thus uniform in $L\geqslant 1/m$).

Our next theorem is a (conservative) Kawasaki version of the previous result. We thus consider the measure $\nu_{\varepsilon,L}^{0}$ obtained by constraining the mean spin of the measure $\nu_{\varepsilon,L}$ to $\sum_{x\in\Omega_{\varepsilon,L}}\varphi_{x}=0$, i.e., $\nu_{\varepsilon,L}^{0}$ is supported on $\{\varphi:\sum_{x}\varphi_{x}=0\}$. (The same proof also works for arbitrary nonzero mean of $\varphi$.) The Dirichlet form for Kawasaki dynamics with invariant measure $\nu_{\varepsilon,L}^{0}$ is defined by

[TABLE]

Theorem 1.7**.**

Fix $\beta<6\pi$, and let $L>0$, $m>0$, and $z\in\mathbb{R}$. Then there is $\gamma^{0}(\beta,z,m,L)>0$ independent of $\varepsilon>0$ such that, for all $F\geqslant 0$,

[TABLE]

Moreover, there is $\delta_{\beta}>0$ such that if $Lm\geqslant 1$ and $|z|m^{-2+\beta/4\pi}\leqslant\delta_{\beta}$, then

[TABLE]

where the constant $O_{\beta}$ depends on $\beta$ only (and is thus uniform in $L\geqslant 1/m$).

For $z=0$, the sine-Gordon model degenerates simply to the continuum Gaussian free field with covariance $(-\Delta+m^{2})^{-1}$, as $\varepsilon\downarrow 0$, for which the Glauber Log-Sobolev constant is $m^{2}$ (by [32] or the Bakry–Émery criterion), and similarly in the Kawasaki case. Note that, in this scaling in which the convexity of the Gaussian measure is of order $1$, the best lower bound on the Hessian of the interaction term $V_{0}$ is of order $-\varepsilon^{-\beta/4\pi}$ if $z\neq 0$ and thus tends to $-\infty$ as $\varepsilon\to 0$. Thus the measure is far out of the scope of the applicability of the Bakry–Émery criterion if $z\neq 0$. Our proof of the above theorems via Theorem 1.2 relies on the smoothing of the effective potential $V_{t}$ along the flow of the Polchinski equation.

The Glauber dynamics of the sine-Gordon model is considered in [36, 16]. Using the theory of regularity structures, it is shown in these references that versions of (1.18) that are regularised in space-time instead of space only converge as $\varepsilon\to 0$ pathwise in a space of distributions on a short noise-dependent time interval. In our setting, it is essential that the noise is white in time for the regularised dynamics to define a Markov process. The question of regularisation in space rather than space-time was considered for the closely related problems of the subcritical continuum $\varphi^{4}$ model and KPZ equation in [34, 35, 66] as well as in [54, 23, 51]. Presumably similar arguments would apply also to the sine-Gordon model, but have not been carried out.

Finally, we provide some references on the continuum sine-Gordon model. For $0<\beta<8\pi$, at least when the domain is a torus and $z\neq 0$ is small and $m^{2}>0$, it is known that $\nu_{\varepsilon,L}\to\nu$ weakly, where $\nu$ is a non-Gaussian measure on $\mathcal{D}^{\prime}(\mathbb{R}^{2})$ with a precise description in terms of renormalised expansions; see [28, 29], [9, 56], [14], and [21, 11, 20] for different approaches. This result is simplest for $\beta<4\pi$, when in finite volume the continuum sine-Gordon measure is absolutely continuous with respect to the Gaussian free field. For $4\pi\leqslant\beta<8\pi$, there is an infinite sequence of thresholds at $\beta=8\pi(1-1/2n)$, $n=1,2,\dots$, at which the partition function (but not the normalised probability measure) acquires divergent contributions; see [9] for further discussion. The physical meaning of these divergences remains debated [27]. The sine-Gordon model satisfies a very interesting duality with the massive Thirring model, the Coleman correspondence or Bosonization [17]. For restricted values of $\beta$, this correspondence has been established in finite volume or with a mass term [29, 18, 8], but in general its proof remains an open problem, most importantly in the formally massless case $m^{2}=0$. In particular, under this correspondence, for the special value $\beta=4\pi$, the correlations functions of the sine-Gordon model are equivalent to those of free fermions. In general, an important question for the sine-Gordon model that has remained open is the formally massless case $L\to\infty$ and $m^{2}\to 0$, in which case correlations decay polynomially if $z=0$. For $z\neq 0$, it is conjectured that the equilibrium correlation functions have exponential decay, for any $\beta<8\pi$. Closely related results for small $\beta$ were obtained in [13, 64]. It would be very interesting to understand the dynamical behaviour in this regime.

Our result extends up to the second threshold $\beta<6\pi$ and makes use of the approach of [14]. It remains a very interesting problem to extend our results to the optimal regime $\beta<8\pi$. Recent progress in the direction of extending the method of [14] includes [43]. Other recent results for the sine-Gordon model include [40]. For a one-dimensional analogue of the sine-Gordon model, a recent construction using martingales was given in [44].

1.4.  More discussion of our approach and of further directions

Our approach to the Log-Sobolev inequality involves the Polchinski equation (1.10). The Polchinski equation is a continuous version of Wilson’s renormalisation group (which typically proceeds in discrete steps) and variations of it go back to [63, 62], while the continuous point of view was first systematically used by Polchinski [59]. See [42] for a review of its history as well as for an account of the important role it has played in recent advances in Perturbative Quantum Field Theory. The relation of the Polchinski equation to the Mayer expansion and its iterated versions was investigated in [14] on which we rely for the sine-Gordon model. Ideas related to the Polchinski equation were also used recently in [5] for a simple construction of the continuum $\varphi^{4}$ model in $d=2,3$. We also mention that approaches involving aspects of renormalisation have been used for a long time to study dynamics of spin systems, e.g., in the form of block dynamics [50, 45, 65] and more recently in the two-scale approach [53, 33, 57, 22]. Our approach involves infinitely many scales.

The regime of the continuum limit considered in Section 1.3 is known as the ultraviolet problem in physics, which for the two-dimensional sine-Gordon model is well-posed for $\beta<8\pi$. The long-distance behaviour is predicted to be independent of $\varepsilon$. For $\beta<8\pi$, it can studied as a property of the continuum limit $\varepsilon\to 0$, but it makes sense for all $\beta>0$ when the regularisation $\varepsilon$ is fixed (lattice problem). For $\beta\geqslant\beta_{c}$ (where the curve $\beta_{c}(z)$ passes through $8\pi$ at $z=0$, see [24, 25]) and small $z$ and $m^{2}=0$, the scaling limit is known to be Gaussian free field in a suitable sense, for the model defined on the torus [25, 19]. This is called the infrared problem in physics. However, we emphasise that, while the ultraviolet problem can be translated to a lattice problem, as we do, the scaling of the infrared problem is more delicate than that of the ultraviolet problem. For the sine-Gordon model, in the ultraviolet limit, the microscopic coupling constant is very small, of order $\varepsilon^{2-\beta/4\pi}\ll 1$. For the infrared problem, the microscopic coupling constant is of order $1$, and unlikely field configurations play a more important role in understanding the measure (large field problem); see [19, 24, 25]. We studied the spectral gap for the hierarchical version of the infrared problem in [6]. Using Theorem 2.6 and the estimates proved in [6], the results for the spectral gap stated in [6] can be improved to results about the Log-Sobolev constant; see Example 2.7.

The next natural class of models that would be interesting to apply Theorem 1.2 to is the $\varphi^{4}$ model. The problem analogous of the one considered for the sine-Gordon model would be the continuum $\varphi^{4}$ model on $\mathbb{R}^{d}$ where $d=2,3$ with sufficiently large mass (ultraviolet problem). On a finite two-dimensional torus, a spectral gap result for the continuum $\varphi^{4}$ model has been shown in [61]. We stress again that the Polchinski equation has also been used in [5] in the construction of the continuum $\varphi^{4}$ model on a torus in $d=2,3$. As in the case of the sine-Gordon model, the infrared problem appears more difficult than the ultraviolet problem. For the latter we expect that the Log-Sobolev constant of the lattice $\varphi^{4}$ model or the Ising model in $d=4$ (respectively $d>4$) scales as $u(-\log u)^{z}$ (respectively $u$) as the critical point is approached with distance $u\downarrow 0$. Again, for the hierarchical $\varphi^{4}$ model, we proved the analogous statement for the spectral gap in [6] and the results of this paper can again be used to improve the latter result to prove also an analogous Log-Sobolev inequality; again see Example 2.7.

In a different direction, the Bakry–Émery theory has a well-known formulation in the context of manifolds (and beyond). The Polchinski equation is closely related to the Gaussian convolution semigroup $\bm{E}_{C_{t}}$ on $X$ and thus to the linear structure of $X$. However with the disintegration of the Gaussian measure taking the role of the reverse Ricci flow, there is an interesting resemblence of our construction with those in [58, 52, 48]; see also Remark 1.5.

Finally, we remark that Log-Sobolev inequalities are a very useful tool to derive mixing results in general, see, e.g., [49]. It would be very interesting to derive such results in our context.

2 Log-Sobolev inequality and the Polchinski equation

In this section we prove Theorem 1.2 and variations of this result that apply in slightly different set-ups. The proofs share many elements with the Bakry–Émery argument which we will review.

2.1.  The renormalisation semigroup

Let $t\in[0,\infty]\mapsto C_{t}$ be a function of positive semidefinite matrices on $\mathbb{R}^{N}$ increasing continuously as quadratic forms to a matrix $C_{\infty}$. More precisely, we assume that $C_{t}=\int_{0}^{t}\dot{C}_{s}\,ds$ for all $t$, where $t\mapsto\dot{C}_{t}$ is a bounded function with values in the space of positive semidefinite matrices that is the derivative of $C_{t}$ except at isolated points. As before, we denote by $\bm{E}_{C_{t}}$ the expectation of the possibly degenerate Gaussian measure with covariance $C_{t}$. We consider a probability measure $\nu_{0}$ with expectation

[TABLE]

for a potential $V_{0}:\mathbb{R}^{N}\to\mathbb{R}$. For $t>s>0$, we recall the definitions

[TABLE]

where the expectations again apply to $\zeta$. We impose the following continuity assumption: For all bounded smooth functions $F:X\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$,

[TABLE]

The assumption (2.5) reduces to (1.6) when $C_{t}$ is differentiable in $t$, as in Section 1.2, and it is again clear in all examples of practical interest.

The following proposition collects some properties of the above definitions; we postpone its elementary proof to Section 2.4.

Proposition 2.1**.**

Let $(C_{t})$ be as above, let $V_{0}\in C^{2}$, and assume (2.5). Then for every $t$ such that $C_{t}$ is differentiable the renormalised potential $V$ defined in (1.3) satisfies the Polchinski equation

[TABLE]

The operators $(\bm{P}_{s,t})_{s\leqslant t}$ form a time-dependent Markov semigroup with generators $(\bm{L}_{t})$, in the sense that $\bm{P}_{t,t}=\mspace{2.0mu}\mathrm{i}\mspace{-0.6mu}\mathrm{d}$ and $\bm{P}_{r,t}\bm{P}_{s,r}=\bm{P}_{s,t}$ for all $s\leqslant r\leqslant t$, that $\bm{P}_{s,t}F\geqslant 0$ if $F\geqslant 0$ and $\bm{P}_{s,t}1=1$, and that for all $t$ at which $C_{t}$ is differentiable (respectively $s$ at which $C_{s}$ is differentiable),

[TABLE]

for all smooth functions $F$, where $\bm{L}_{t}$ acts on a smooth function $F$ by

[TABLE]

The measures $\nu_{t}$ evolve dual to $(\bm{P}_{s,t})$ in the sense that

[TABLE]

Finally, for any smooth function $F$ with values in a compact subset of $(0,\infty)$ and $\Phi(x)=x\log x$,

[TABLE]

Remark 2.2**.**

The Polchinski semigroup operates from the right, i.e., $\bm{P}_{s,t}=\bm{P}_{r,t}\bm{P}_{s,r}$ for $s\leqslant r\leqslant t$. Thus it acts on probability densities relative to $\nu_{t}$: if $\mu_{0}=F\,d\nu_{0}$ is a probability measure then $\mu_{t}=\bm{P}_{0,t}F\,d\nu_{t}$ is again a probability measure. For a time-independent semigroup $\bm{T}_{s,t}=\bm{T}_{t-s}$ that is reversible with respect to the measure $\nu_{0}$ (as, for example, the original semigroup associated to the Dirichlet form), one has the dual point of view that $\bm{T}$ describes the evolution of an observable:

[TABLE]

Such a dual semigroup can be realised in terms of a Markov process $(\varphi_{t})$ as $\bm{T}_{t}F(\varphi)=\bm{E}_{\varphi_{0}=\varphi}F(\varphi_{t})$. Since the Polchinski semigroup is not reversible and time-dependent, this interpretation does not apply to the Polchinski semigroup. Instead, the Polchinski semigroup $\bm{P}_{s,t}$ can be realised in terms of an SDE that starts at time $t$ and runs time in the negative direction from $t$ to $s$. Indeed, set $\varphi_{r}=\tilde{\varphi}_{t-r}$ where $\tilde{\varphi}$ satisfies

[TABLE]

Since $G(r,\varphi)=\bm{P}_{s,t-r}F(\varphi)$ satisfies $\partial_{r}G+\bm{L}_{t-r}G=0$ for $s<r<t$ by (2.7), Itô’s formula and (2.12) imply that $G(r,\tilde{\varphi}_{r})=\bm{P}_{s,t-r}F(\varphi_{t-r})$ is a martingale for $r\in[s,t]$. This implies

[TABLE]

Thus if $\varphi_{t}$ is distributed according to $\nu_{t}$ by the above backward in time evolution $\varphi_{s}$ is distributed according to $\nu_{s}$ for $s<t$. Our interpretation of this is that, while the renormalised measures $\nu_{t}$ are supported on increasing smooth (in the index space) configurations as $t$ grows, the backward evolution restores the small scale fluctuations of $\nu_{0}$.

For later use we also record the following useful relations for the derivatives of $V_{t}$; we will not use these in Section 2. The formulas follow immediately by differentiating (2.2) using (2.3).

Proposition 2.3**.**

For all $f\in X$ and $t\geqslant s\geqslant 0$,

[TABLE]

2.2.  Relative entropy, Markov semigroups, and the Bakry–Émery argument

In a time-dependent generalisation, we now review the decomposition of the relative entropy in terms of a semigroup that underlies the Bakry–Émery argument. By approximation (see, e.g., [60, Theorem 3.1.13]), to prove a Log-Sobolev inequality, it suffices to consider smooth functions $F:X\to\mathbb{R}$ with values in a compact subset of $(0,\infty)$, which we will do from now on.

We consider a curve of probability measures $(\nu_{t})_{t\geqslant 0}$ and a corresponding dual time-dependent Markov semigroup $(\bm{P}_{s,t})$ with generators $(\bm{L}_{t})$ as in Proposition 2.1. Namely, we assume that (2.7) and (2.9) hold, that $\bm{L}_{t}$ is of the form (2.8) for some positive semidefinite matrices $\dot{C}_{t}$ and functions $V_{t}$ (not necessarily satisfying (2.6)), and also that (2.10) holds. Denoting $F_{t}=\bm{P}_{0,t}F$ and $\dot{F}_{t}=\frac{\partial}{\partial t}F_{t}$, using first (2.9) and then (2.8), it is then elementary to see that

[TABLE]

Integrating this relation using (2.10), with $\Phi^{\prime\prime}(x)=1/x$, it follows that

[TABLE]

To be precise, recall that $C_{t}$ is differentiable except for at most countably many $t$. For all $t$ such that $C_{t}$ is differentiable, the identity (2.2) holds and implies that the continuous function $t\mapsto\mathbb{E}_{\nu_{t}}\Phi(F_{t})$ is differentiable at $t$ with nonpositive derivative. In particular, this implies that $\mathbb{E}_{\nu_{t}}\Phi(F_{t})$ is decreasing, which justifies the use of the fundamental theorem of calculus and together with (2.5) with $t=+\infty$ for the limit gives (2.17).

To obtain a Log-Sobolev inequality, the right-hand side of (2.17) must be bounded by the Dirichlet form with respect to the measure $\nu_{0}$. The same argument with $\Phi(x)=x^{2}$ would give a bound on the variance rather than the entropy and correspondingly a spectral gap inequality; the required bound is easier to obtain in this case.

For measures that are log-concave (or, more generally, ones that satisfy a curvature dimension condition; see [4]), sharp estimates have been obtained by celebrated arguments of Lichnerowicz (for the spectral gap) and of Bakry–Émery. We review the latter briefly now.

Example 2.4** (Bakry–Émery [3, 2]).**

Assume the measure $\nu=\nu_{0}$ has expectation given by (1.12). Let $\nu_{t}=\nu_{0}$ for all $t\geqslant 0$, and define the semigroup $\bm{T}_{s,t}=\bm{T}_{t-s}$ with generator

[TABLE]

This semigroup leaves $\nu_{0}$ invariant. Bakry–Émery showed, for all $F\geqslant 0$,

[TABLE]

If $\operatorname{Hess}H(\varphi)\geqslant\lambda\mspace{2.0mu}\mathrm{i}\mspace{-0.6mu}\mathrm{d}>0$ as quadratic forms, uniformly in $\varphi\in\mathbb{R}^{N}$, it follows that

[TABLE]

Substituting this into (2.17) yields the Log-Sobolev inequality

[TABLE]

In fact, (2.4) follows as in Lemma 2.8 below.

2.3.  Variations of Theorem 1.2

The following theorem generalises Theorem 1.2 by not assuming that $\dot{C}_{t}$ is given by the heat kernel.

Theorem 2.5**.**

Let $\dot{C}_{t}$ and $V_{t}$ be as in Section 2.1, assume that $\dot{C}_{t}$ is differentiable for all $t$, and that (2.5) holds. Suppose there are $\dot{\lambda}_{t}$ (allowed to be negative) such that

[TABLE]

and define

[TABLE]

where $|\dot{C}_{0}|$ is the largest eigenvalue of $\dot{C}_{0}$. Then $\nu_{0}$ satisfies the Log-Sobolev inequality

[TABLE]

The proof of the theorem is given in Section 2.5. When $\dot{C}_{t}$ is given by the heat kernel, as in the context of Theorem 1.2, the term $\ddot{C}_{t}$ in (2.22) can be eliminated explicitly and we can thus deduce Theorem 1.2 as follows.

Proof of Theorem 1.2.

Let $Q_{t}=e^{-tA/2}$ and $\dot{C}_{t}=e^{-tA}=Q_{t}^{2}$. Then $\ddot{C}_{t}=-A\dot{C}_{t}=-Q_{t}AQ_{t}$ and the left-hand side of (2.22) is equal to

[TABLE]

Since by assumption $A\geqslant\lambda$ and $Q_{t}\operatorname{Hess}V_{t}Q_{t}\geqslant\dot{\mu}_{t}$ we can choose $\dot{\lambda}_{t}=\frac{1}{2}\lambda+\dot{\mu}_{t}$ to get

[TABLE]

which with $Q_{t}^{2}=\dot{C}_{t}$ implies (2.22). The claim (1.9) is thus implied by Theorem 2.5. ∎

The next theorem provides a variation of Theorem 2.5 that does not rely on differentiability or even continuity of $\dot{C}_{t}$ in $t$, and can therefore be applied with more general covariance decompositions. The price is the less symmetric condition (2.27). However, this condition can for example be applied to discrete decompositions $C_{\infty}=C_{0}+C_{1}+\cdots$ by setting $\dot{C}_{s}=\sum_{j}1_{(j,j+1]}(s)C_{j}$. In particular, this applies to the hierarchical spin models that we studied in [6]; see Example 2.7.

Theorem 2.6**.**

Let $\dot{C}_{t}$ and $V_{t}$ be as in Section 2.1, and let $X_{t}\subseteq X$ be the image of the matrix $C_{\infty}-C_{t}$. Assume that (2.5) holds and that there are $\dot{\lambda}_{t}$ (allowed to be negative) such that

[TABLE]

and define

[TABLE]

where $|\dot{C}_{t}|$ is the largest eigenvalue of $\dot{C}_{t}$. Then $\nu_{0}$ satisfies the Log-Sobolev inequality (2.24).

Again the proof is given in Section 2.5.

Example 2.7** (Hierarchical models).**

Let $C_{j}=\mu_{j}Q_{j}$ be the decomposition of the hierarchical Green function as in [6, Section 2.1] (where we here write $\mu_{j}$ instead of $\lambda_{j}$) and set $\dot{C}_{t}=\sum_{j}1_{(j,j+1]}(t)C_{j}$ and $\dot{Q}_{t}=\sum_{j}1_{(j,j+1]}(t)Q_{j}$. Using the structure of the hierarchical decomposition, for $\varphi\in X_{t}$, the matrix $\operatorname{Hess}V_{t}(\varphi)$ is block diagonal with respect to scale-$j$ blocks (see [6, Section 1.3]) where $t\in(j,j+1]$ and constant on each such block. This means that $\operatorname{Hess}V_{t}(\varphi)$ commutes with $Q_{t}$ and by the hierarchical structure thus with $\dot{C}_{t}$. In particular, for $\varphi\in X_{t}$,

[TABLE]

implies (2.27). For hierarchical versions of the four-dimensional lattice $|\varphi|^{4}$ model in the approach of the critical point, and for the two-dimensional lattice sine-Gordon model in the rough (Kosterlitz–Thouless) phase, we established the estimate (2.29) for integer $t$ (and appropriate $\dot{\lambda}_{t}$) in [6]. By the same methods, one can extend those estimates to noninteger $t$ with $-\dot{\lambda}_{t}=O(-\dot{\lambda}_{j})$ for $t\in(j,j+1]$. Using Theorem 2.6 instead of [6, Theorem 2.1], the theorems for the spectral gap in [6] can thus be extended to analogous ones for the Log-Sobolev constant.

Further variations of the conditions (2.22) and (2.27) for the Log-Sobolev inequality are possible and might be useful in other applications, but we do not investigate these here.

2.4.  Proof of Proposition 2.1

We start with the proof of Proposition 2.1. This is a straightforward computation from the definitions.

Proof of Proposition 2.1.

Let $Z_{t}(\varphi)=\bm{E}_{C_{t}}e^{-V_{0}(\varphi+\zeta)}$. By a well-known computation (see, e.g., [7, Section 2]), it follows that the Gaussian convolution acts as the heat semigroup with time-dependent generator $\frac{1}{2}\Delta_{\dot{C}_{t}}$, i.e., if $Z_{0}$ is $C^{2}$ in $\varphi$ so is $Z_{t}$ for any $t>0$, that $Z_{t}(\varphi)>0$ for any $t$ and $\varphi$, and that for any $t>0$ such that $C_{t}$ is differentiable,

[TABLE]

Therefore $V_{t}=-\log Z_{t}$ satisfies the Polchinski equation

[TABLE]

That $(\bm{P}_{s,t})$ is a semigroup, i.e., that $\bm{P}_{r,t}\bm{P}_{s,r}=\bm{P}_{s,t}$ and $\bm{P}_{t,t}=\mspace{2.0mu}\mathrm{i}\mspace{-0.6mu}\mathrm{d}$ for any $s\leqslant r\leqslant t$, follows immediately from the definition (1.4) and the convolution property of Gaussian measures, i.e., that the sum of two independent Gaussian vectors is Gaussian with covariance given by the sum of the covariances (again see, e.g., [7, Section 2]). The Markov property is obvious. To verify that its generator $\bm{L}_{t}$ is given by (2.8), set $F_{t}(\varphi)=\bm{P}_{0,t}F(\varphi)=e^{V_{t}(\varphi)}\bm{E}_{C_{t}}(e^{-V_{0}(\varphi+\zeta)}F(\varphi+\zeta))$. Then

[TABLE]

which is the second equality in (2.7). The third inequality in (2.7) follows analogously, and the first inequality is clear from the fact that the Gaussian measure with covariance [math] is the Dirac measure at [math].

The first equality in (2.9) holds by definition, and the second one is a direct computation from the definition (1.3) and the fact that $V$ satisfies (1.10):

[TABLE]

Finally, (2.10) follows from (2.5). Indeed, if $F$ takes values in a compact interval $I\subset(0,\infty)$, then $\bm{P}_{0,t}F$ also takes values in $I$. The function $\Phi$ is smooth on $I$ and can be extended to a bounded smooth function $g$ on $\mathbb{R}$ such that $g|_{I}=\Phi|_{I}$. The claim now follows from (2.5). ∎

2.5.  Proofs of Theorems 2.5-2.6

Theorems 2.5-2.6 can be proved in the same way as the Bakry–Émery criterion with the crucial difference that the original semigroup is replaced by the Polchinski semigroup, that the corresponding potentials depend on time, and that gradients are taken in terms of a time-dependent quadratic form. We present the primary proofs along the lines of [4]; see Remark 2.9 for alternative proofs using synchronous coupling as in [15].

Lemma 2.8**.**

Let $\bm{L}_{t}$, $\bm{P}_{0,t}$, $\dot{C}_{t}$, $V_{t}$ be as in Section 2.1. Then the following identity holds for any $t$-independent positive definite matrix $Q$:

[TABLE]

where $|M|_{2}^{2}=\sum_{p,q}|M_{pq}|^{2}$ denotes the squared Frobenius norm of a matrix $M=(M_{pq})$.

Proof.

Throughout the proof, we drop the fixed index $t$, i.e., write $F$ instead of $\bm{P}_{0,t}F$, and $\bm{L}$ for $\bm{L}_{t}$, and similarly for $\dot{C}_{t}$ and $V_{t}$. Then the left-hand side of (2.34) can be written as

[TABLE]

To compute the three terms, we denote derivatives by subscripts $i,j,k,l$, and use the summation convention for these subscripts. The first term then is

[TABLE]

where the last bracket can be expanded as

[TABLE]

The sum of the second and third terms in (2.35) is

[TABLE]

By adding all three terms, we obtain that (2.35) equals

[TABLE]

Using that for any given indices $i,j,k,l$,

[TABLE]

equation (2.39) can be written as

[TABLE]

Using that $2(\sqrt{F})_{j}=F_{j}/\sqrt{F}$ for the first term, and that, for any symmetric matrix $M$,

[TABLE]

for the second term, (2.41) can therefore be written as

[TABLE]

Proof of Theorem 2.5.

Lemma 2.8 with $Q=\dot{C}_{s}$ implies

[TABLE]

By the assumption (2.22) and since the last term is positive, it follows that

[TABLE]

Equivalently, $\psi(s):=e^{-2\lambda_{t}+2\lambda_{s}}\bm{P}_{s,t}\left[(\nabla\sqrt{\bm{P}_{0,s}F})^{2}_{\dot{C}_{s}}\right]$ satisfies $\psi^{\prime}(s)\leqslant 0$ for $s<t$. This implies

[TABLE]

By (2.17), then (2.24) follows. ∎

Proof of Theorem 2.6.

Lemma 2.8 with $Q=\mspace{2.0mu}\mathrm{i}\mspace{-0.6mu}\mathrm{d}$ implies

[TABLE]

By the assumption (2.27) and since the last term is positive, it follows that, on $X_{s}$,

[TABLE]

Equivalently, pointwise on $X_{t}$, $\psi(s):=e^{-2\lambda_{t}+2\lambda_{s}}\bm{P}_{s,t}\left[(\nabla\sqrt{\bm{P}_{0,s}F})^{2}\right]$ satisfies $\psi^{\prime}(s)\leqslant 0$ for $s<t$. This implies, on $X_{t}$,

[TABLE]

Again by (2.17), using that $\nu_{t}$ is supported on $X_{t}$, (2.24) follows. ∎

Remark 2.9**.**

Using the representation (2.12)-(2.13) of the semigroup $\bm{P}_{s,t}$ in terms of a stochastic process (that evolves backwards in time from $t$ to $s$), one can alternatively prove the theorems using synchronous coupling as in [15].

3 Application to the continuum sine-Gordon model

In this section, we prove Theorems 1.6 and 1.7 by applying Theorem 1.2. While it is not necessary, we find it clearest to rescale the continuum sine-Gordon model at scale $\varepsilon$ to a unit lattice problem.

3.1.  Rescaling and heat kernel decomposition

Identifying $\Omega_{\varepsilon,L}$ with the unit lattice $\Lambda=\frac{1}{\varepsilon}\Omega_{\varepsilon,L}$, the continuum sine-Gordon model $\nu_{\varepsilon,L}$ is equivalent to a spin system whose coupling matrix is given by the nearest neighbour Laplacian on $\mathbb{Z}^{d}$. We will thus drop the subscripts $\varepsilon,L$ now, and write $\nu_{0}$ for the measure of the form (1.1) with $X=\mathbb{R}^{\Lambda}$ and

[TABLE]

where $\Delta_{\Lambda}$ is the standard unit lattice Laplacian acting on the discrete torus of side length $L/\varepsilon$. We emphasise that throughout this section $\Delta_{\Lambda}$ denotes the lattice Laplacian on $\Lambda$ and not the Laplacian on $\mathbb{R}^{\Lambda}$ which we denoted $\Delta_{\dot{C}_{t}}$ in the previous section. Note that $\varphi$ is not rescaled. As is natural in this normalisation, we normalise the Glauber Dirichlet form, for $F:\mathbb{R}^{\Lambda}\to\mathbb{R}$, by

[TABLE]

Note that in this normalisation the Log-Sobolev constant of the non-interacting (Gaussian) model with $z=0$ scales as $\varepsilon^{2}m^{2}$ (corresponding to the unit order Log-Sobolev constant $m^{2}>0$ in the continuum scaling). Also note that the correlation length of the non-interacting model scales as $1/(m\varepsilon)$, making it natural to assume $L\geqslant 1/m$ as in the statements of the theorems.

In the following, we will use Theorem 1.2 to prove the same scaling in $\varepsilon$ for the Log-Sobolev constant of the interacting model. To verify the assumptions of Theorem 1.2, we will prove the following estimates on $V_{t}$ as defined in (1.3). We recall that $Q_{t}=e^{-tA/2}$ denotes the heat kernel on the index space $\Lambda$.

Proposition 3.1**.**

Let $\beta<6\pi$, and $L>0$, $m>0$, and $z\in\mathbb{R}$. Then (1.6) holds, and for all $t\geqslant 0$,

[TABLE]

where $\mu_{t}=\int_{0}^{t}\dot{\mu}_{s}\,ds$ satisfies

[TABLE]

with $\mu^{*}=\mu^{*}(\beta,z,m,L)$ independent of $\varepsilon>0$. Moreover, there is $\delta_{\beta}>0$ such that if

[TABLE]

then the optimal bound satisfies $\mu^{*}=O_{\beta}(|z|m^{-2+\beta/4\pi})$ uniformly in $L$.

Indeed, Theorem 1.6 is an immediate consequence of these estimates and Theorem 1.2.

Proof of Theorem 1.6.

The smallest eigenvalue of $A$ is $\lambda=\varepsilon^{2}m^{2}$. By (1.9) and (3.4), therefore

[TABLE]

and Theorem 1.2 implies that $\nu_{0}$ satisfies a Log-Sobolev inequality with constant $\gamma$. In the continuum normalisation of the Dirichlet form (1.17), the sine-Gordon measure thus satisfies a Log-Sobolev inequality with constant given by $m^{2}e^{-2\mu^{*}}$. Moreover, if (3.5) holds, then $m^{2}e^{-2\mu^{*}}=m^{2}+O_{\beta}(m^{\beta/4\pi}|z|)$. ∎

The proof of Theorem 1.7 for Kawasaki dynamics is almost the same as that of Theorem 1.6. The constraint measure $\nu_{0}^{0}$ can be written as in (2.1), with the degenerate covariance matrix $C_{\infty}^{0}$ supported on the subspace $X=\mathbb{R}^{\Lambda}_{0}=\{\varphi\in\mathbb{R}^{\Lambda}:\sum_{x}\varphi_{x}=0\}$ given by

[TABLE]

In unit lattice scaling, the Dirichlet form for Kawasaki dynamics is given, for $F:\mathbb{R}^{\Lambda}_{0}\to\mathbb{R}$, by

[TABLE]

We decompose the covariance matrix $C_{\infty}^{0}$ in terms of

[TABLE]

and define $V_{t}^{0}$ as in (1.3) with respect to $\dot{C}_{t}^{0}$. From now on, we will refer to the case that $V_{t}$ is replaced by $V_{t}^{0}$ and $\dot{C}_{t}$ by $\dot{C}_{t}^{0}$ as the conservative case. Then the statement of Proposition 3.1 remains true in the conservative case.

Proposition 3.2**.**

Let $\beta<6\pi$, and $L>0$, $m>0$, and $z\in\mathbb{R}$. Then (1.6) holds, and for all $t\geqslant 0$,

[TABLE]

where $\mu_{t}$ satisfies (3.4) with the same bound on $\mu^{*}$ if (3.5) holds.

Analogously as in the proof of Theorem 1.6, we deduce Theorem 1.7 from Proposition 3.2.

Proof of Theorem 1.7.

Since $\Lambda$ is a discrete torus of side length $L/\varepsilon$, the smallest nonzero eigenvalue of the lattice Laplacian $-\Delta_{\Lambda}$ on $\Lambda$ is of order $(\varepsilon/L)^{2}$. We thus denote the smallest nonzero eigenvalue of $-\Delta_{\Lambda}$ on $\Lambda$ by $\zeta^{2}\varepsilon^{2}$. Explicitly, as $\varepsilon\to 0$,

[TABLE]

As in the proof of Theorem 1.6, with $\lambda$ the smallest eigenvalue on $X$ of $A=-\Delta_{\Lambda}+\varepsilon^{2}m^{2}$,

[TABLE]

and Theorem 1.2 implies that $\nu_{0}^{0}$ satisfies a Log-Sobolev inequality with constant $\gamma^{0}$:

[TABLE]

where the last inequality again uses that the smallest nonzero eigenvalue of the lattice Laplacian $-\Delta$ is $\varepsilon^{2}\zeta^{2}$. We emphasise that $\nabla$ denotes the continuous gradient on $\mathbb{R}^{\Lambda}$ while $\Delta_{\Lambda}$ is the lattice Laplacian on $\Lambda$. Recalling the continuum normalisation of the Dirichlet form given by (1.21), and (3.4), this is the claim of Theorem 1.7. ∎

3.2.  Outline, scaling conventions, and heat kernel

To prove Propositions 3.1-3.2, we proceed in the following steps. We first consider the main case (3.5). The proofs are simpler for $\beta<4\pi$ and we begin with this case in Section 3.4. In Sections 3.5-3.7, we extend this analysis to the case $\beta<6\pi$. Finally, in Section 3.8, we show that a crude argument suffices to remove the assumption (3.5) at the cost of constants that are uniform in $\varepsilon$ but not in $L$.

To prove Propositions 3.1-3.2, we will require estimates on the heat kernel decomposition

[TABLE]

In this section, we set-up a convenient normalisation and also collect some elementary estimates. We have chosen the heat kernel decomposition (and not a finite range decomposition, for example) to be able to directly apply Theorem 1.2. The characteristic length scale of the heat kernel is defined by

[TABLE]

and we set

[TABLE]

Standard estimates on the heat kernel imply that $\dot{C}_{t}(x,y)$ is essentially supported on $|x-y|\lesssim\ell_{t}$ and the above normalisation is such that $\mathsf{\dot{C}}_{\lambda^{2}t}(\lambda x,\lambda y)\approx\mathsf{\dot{C}}_{t}(x,y)$ and $\mathsf{Q}_{t}^{2}=\mathsf{\dot{C}}_{t}$. We will often express estimates in terms of these quantities and in terms of $\ell_{t}$ (instead of $t$), and write integrals over the scale in terms of the approximately scale invariant measure $dt/\ell_{t}^{2}\approx dt/t$ (instead of $dt$). For estimates involving the heat kernels $Q_{t},\dot{C}_{t},C_{t}$ and its scaled versions, we will always impose the following assumption:

[TABLE]

The next lemma provides some elementary estimates on the heat kernel. These are sufficient for the case $\beta<4\pi$; for $\beta>4\pi$ more precise estimates are required (and will be stated in the section they are used). All of these estimates on the heat kernel are collected in Appendix A.

Lemma 3.3**.**

Assume (3.17). For any $x\in\Lambda$,

[TABLE]

and the same estimates hold in the conservative case.

Proof.

This follows from standard estimates on the heat kernel on $\mathbb{Z}^{2}$, see Appendix A. ∎

Further we define the scale dependent coupling constant $\mathsf{z}_{t}$ and its microscopic version $z_{t}$ by

[TABLE]

For later purposes, we will now collect some basic properties of this definition. By (3.18) and the definitions of $\mathsf{z}_{t}$ and $\ell_{t}$, uniformly in $t>0$,

[TABLE]

In the following, we write $x\lesssim y$ or $x=O_{\beta}(y)$ if $|x|\leqslant C_{\beta}|y|$ for a $\beta$-dependent constant $C_{\beta}$. For any $\beta<8\pi$, by (3.20) then

[TABLE]

as is straightforward to check from the definitions. For use in the proof for $\beta>4\pi$, we also record the following estimates (again straightforward from the definitions): for all positive integers $n$,

[TABLE]

3.3.  Fourier representation

To estimate the Hessian of the renormalised potential $V_{t}$, we use the Brydges–Kennedy approach [14]. Namely, for any function $V:\mathbb{R}^{\Lambda}\to\mathbb{R}$ that is $\frac{2\pi}{\sqrt{\beta}}$-periodic in each variable, we will write its Fourier series (assuming it converges absolutely) as

[TABLE]

where $\tilde{V}^{(n)}:(\Lambda\times\{\pm 1\})^{n}\to\mathbb{R}$ and

[TABLE]

We think of $\xi_{i}$ as a particle with position $x_{i}$ and charge $\sigma_{i}$. Since the index $n$ is determined from the number of arguments of $\tilde{V}^{(n)}$, we will often omit it and write $\tilde{V}(\xi_{1},\dots,\xi_{n})=\tilde{V}^{(n)}(\xi_{1},\dots,\xi_{n})$. The representation (3.24) is not manifestly unique without further conditions, but in the relevant cases we will in fact construct coefficients $\tilde{V}(\xi_{1},\dots,\xi_{n})$ such that (3.24) holds.

The initial potential $V_{0}$ of the sine-Gordon model corresponds to

[TABLE]

Set

[TABLE]

and

[TABLE]

where $[n]=\{1,\dots,n\}$. We define $u_{s}$ and $W_{s}$ analogously by replacing $\dot{C}_{s}$ by $C_{s}$. For later use, we note that $W_{t}-W_{s}\geqslant 0$ holds for all arguments by positive definiteness of $\dot{C}_{s}$.

Then in terms of the Fourier representation (3.24), the two terms on the right-hand side of the Polchinski equation (1.10) are represented by

[TABLE]

The sum over $I_{1}\dot{\cup}I_{2}=[n]$ is over all nonempty disjoint subsets $I_{1}$ and $I_{2}$ of $[n]$ with $I_{1}\cup I_{2}=[n]$. Moreover, given $\xi_{1},\dots,\xi_{n}$ and $I=\{i_{1},\dots,i_{k}\}\subset[n]$ we denote by $\xi_{I}$ the vector $(\xi_{i_{1}},\dots,\xi_{i_{k}})$.

Indeed, (3.3) is straightforward to verify in the sense that if $V$ is given by (3.24) and $\widetilde{\Delta_{\dot{C}_{s}}V}$ by (3.3) then

[TABLE]

To see (3.30), note that differentiating (3.24) gives

[TABLE]

and thus

[TABLE]

Therefore taking the sum over $p$ and $q$, using that the number partitions of $[n]$ into two subsets with $p$ and $q=n-p$ elements is $n!/(p!q!)$ and that $\tilde{V}$ is symmetric in its arguments, we find

[TABLE]

if $\widetilde{(\nabla V,\nabla V)}_{\dot{C}_{s}}$ is given by (3.30).

By (3.3)-(3.30) and the Duhamel principle, the Polchinski equation has the following formulation as an integral equation:

[TABLE]

For $n\leqslant 1$, the unique solution to (3.35) is simply

[TABLE]

with $z_{t}$ defined in (3.19). For $n>1$, $\tilde{V}_{t}(\xi_{1},\dots,\xi_{n})$ is then determined explicitly by (3.35) in terms of $\tilde{V}_{s}(\xi_{1},\dots,\xi_{k})$, $k<n$. Hence by induction, (3.35) has a unique solution for any $n$ and $t$. This is summarised in the following lemma along with a uniqueness property.

Lemma 3.4**.**

The integral equation (3.35) has a unique solution $\tilde{V}$ for all $n$ and $t$. Moreover, if $V_{t}$ defined in terms of $\tilde{V}_{t}$ by (3.24) converges absolutely, locally uniformly in $t>0$, then $V_{t}$ is equal to (1.3), the convolution solution of the Polchinski equation.

Proof.

We have already shown that (3.35) has a unique solution. For coefficients $\tilde{V}_{t}$ such that (3.24) and its derivatives converge absolutely, the function $V_{t}$ defined by (3.24) is smooth. Moreover, for smooth $V_{t}$, the integral equation (3.35) implies the Polchinski equation (1.10). Uniqueness of bounded solutions to the Polchinski equation by Remark 1.4 then implies that $V_{t}$ coincides with the convolution solution of the Polchinski equation. ∎

3.4.  Up to the first threshold: proof of Propositions 3.1-3.2 for $\beta<4\pi$ assuming (3.5)

The following proposition, due to [14], gives good bounds when $\beta<4\pi$. For completeness, we reproduce their argument here in our set-up and notation. (See also [30, 31, 12, 38, 43] for related results.) We will then use the result to derive Proposition 3.1 in the case $\beta<4\pi$. Let

[TABLE]

and

[TABLE]

Proposition 3.5**.**

For all $n\geqslant 1$, the solution to (3.35) satisfies

[TABLE]

with $z_{t}$ defined in (3.19). In particular, if $z_{t}M_{t}<1/e$, the Fourier series for $V_{t}$ converges and $V_{t}$ coincides with the convolution solution to the Polchinski equation. The analogous statements hold in the conservative case.

Proof.

For $n=1$, the bound (3.39) is obvious from (3.36). To prove the bounds (3.39) for $n>1$, we use induction. Note that the first term on the right-hand side of (3.35) does not contribute for $n>1$ since then $\tilde{V}_{0}^{(n)}=0$ by (3.26). In the second term, we drop the exponential inside the integral (as $W_{t}-W_{s}\geqslant 0$) to obtain

[TABLE]

Note that if $|I_{1}|=n-k$ and $|I_{2}|=k$ then

[TABLE]

For example,

[TABLE]

Assuming the bound (3.39) for integers less than $n$, therefore

[TABLE]

Using that $\sum_{k=1}^{n-1}\binom{n}{k}k^{k-1}(n-k)^{n-k-1}=2(n-1)n^{n-2}$ and $n/2\leqslant n-1$ for $n\geqslant 2$,

[TABLE]

For $n>2$, the last equality follows from the following change of variables,

[TABLE]

applied with $g(s)=\|\dot{u}_{s}\|e^{-\beta C_{s}(0,0)}$. Indeed,

[TABLE]

Finally, using the bounds (3.39) for $\tilde{V}_{t}(\xi_{1},\dots,\xi_{n})$ and the assumption $\sup_{t}z_{t}M_{t}<1/e$, the series (3.24) for $V_{t}(\varphi)$ converges absolutely since (using $n^{n}/n!\leqslant e^{n}$),

[TABLE]

and analogously for derivatives. Hence $V$ solves the Polchinski equation (1.10) by Lemma 3.4. ∎

Using the conclusion of the last proposition together with the basic estimates for $\mathsf{\dot{C}}_{s}$ given in Lemma 3.3, it is straightforward to complete the proof of Propositions 3.1-3.2 for $\beta<4\pi$.

Proof of Propositions 3.1-3.2 for $\beta<4\pi$ assuming (3.5).

Since the proofs of the two propositions are identical we only discuss Proposition 3.1. From (3.18),

[TABLE]

For $\beta<4\pi$, the definition of $M_{t}$ in (3.39), the definition of $\ell_{t}$ in (3.15), and (3.18) imply

[TABLE]

In this proof, the condition $\beta<4\pi$ is only needed in order to achieve the scaling $\ell_{t}^{2}$ in the previous upper bound. By (3.19)-(3.20) therefore, using in the last inequality that $|z|m^{-2+\beta/4\pi}$ is sufficiently small,

[TABLE]

Let

[TABLE]

From (3.24) together with (3.39), (3.49), and with $n^{n}/n!\leqslant e^{n}$ we obtain

[TABLE]

Since $|(f,\operatorname{Hess}V_{t}(\varphi)f)|\leqslant\|\operatorname{Hess}V_{t}(\varphi)\||f|_{2}^{2}$ and $|Q_{t}f|_{2}\leqslant\vartheta_{t}|f|_{2}$, we obtain

[TABLE]

In the notation of Theorem 1.2 we thus have that $\dot{\mu}_{t}\geqslant-O_{\beta}(|z_{t}|\vartheta_{t}^{2})$. Hence, using the bounds for $z_{t}$ from (3.21) and (3.20), for all $t\geqslant 0$,

[TABLE]

Finally, the ergodicity assumption (1.6) follows from the weak-* convergence $\nu_{t}\to\nu_{\infty}\equiv\delta_{0}$ and $\bm{P}_{0,t}F(\varphi)\to\bm{P}_{0,\infty}F(\varphi)$ uniformly in $\varphi$. Indeed, $\nu_{t}\to\nu_{\infty}$ holds since the Gaussian measure covariance $C_{\infty}-C_{t}$ converges to $\delta_{0}$ and $V_{t}(\varphi)$ is bounded (uniformly in $\varphi$ and $t$). The uniform convergence $\bm{P}_{0,t}F\to\bm{P}_{0,\infty}F$ holds since $V_{t}(\varphi)\to V_{\infty}(\varphi)$ and $\bm{E}_{C_{s}}e^{-V_{0}(\varphi+\zeta)}F(\varphi+\zeta)\to\bm{E}_{C_{\infty}}e^{-V_{0}(\varphi+\zeta)}F(\varphi+\zeta)$, both uniformly in $\varphi$, where the last claim holds since the integrand is a bounded Lipschitz function. ∎

3.5.  Up to the second threshold:

proof of Propositions 3.1-3.2 for $\beta<6\pi$ assuming (3.5)

The remainder of Section 3 is devoted to extending the proof of Proposition 3.1 from $\beta<4\pi$ to $\beta<6\pi$. For this, we will estimate the $n=2,3,4$ terms in (3.24) more carefully.

Indeed, for $n=2$, a uniform bound on $\tilde{V}_{t}(\xi_{1},\xi_{2})$ as used for $\beta<4\pi$ is not true when $\beta\geqslant 4\pi$, and we rely crucially on the smoothing effect of the heat kernel $Q_{t}$ in (1.8) to obtain the required bound stated in the following proposition. (Note that this estimate is best expressed in terms of $\mathsf{Q}_{t}$ and $\mathsf{z}_{t}$ rather than $Q_{t}$ and $z_{t}$.)

Proposition 3.6**.**

Let $\beta<8\pi$ and assume (3.17). Then

[TABLE]

The analogous statement holds in the conservative case.

For the terms $n>2$, the following proposition gives an analogue of Proposition 3.5 for $\beta<6\pi$.

Proposition 3.7**.**

Let $\beta<6\pi$ and assume (3.17). Then there is $C_{\beta}<\infty$ such that for all $n\geqslant 3$,

[TABLE]

The analogous statement holds in the conservative case.

These bounds together imply Propositions 3.1-3.2 when (3.5) holds.

Proof of Propositions 3.1-3.2 assuming (3.5).

Since the proofs are again the same, and we only prove Propositions 3.1. The bound (3.56) (together with the qualitative fact that $V^{(1)}$ and $V^{(2)}$ are finite) implies that (3.24) converges, exactly as in (3.47). Moreover, exactly as in (3.52)-(3.53), for $|z|m^{-2+\beta/4\pi}$ sufficiently small, it follows that

[TABLE]

Combined with (3.55) this gives the required bound (3.3). The proof of the ergodicity assumption (1.6) is also identical to that in the proof of Proposition 3.1 for $\beta<4\pi$. ∎

To prove the above propositions, neutral configurations require more careful treatment compared to the case $\beta<4\pi$, where neutral means the following. For a configuration $\xi=(\xi_{1},\dots,\xi_{k})$ we define the charge $\sigma(\xi)=\sum_{i=1}^{k}\sigma_{i}$ and call $\xi$ neutral if $\sigma(\xi)=0$ and call $\xi$ charged otherwise. We will sometimes decompose

[TABLE]

where $V^{(n,0)}$ is defined as in (3.24) with the sum over $\xi=(\xi_{1},\dots,\xi_{n})$ restricted to neutral $\xi$, and $V^{(n,\pm)}$ by restricting the sum to charged $\xi$. As in the proof for $\beta<4\pi$, the starting point for the proofs is (3.35), but now without dropping the exponential inside the integral, i.e., for $n>1$,

[TABLE]

3.6.  Proof of Proposition 3.6: the term $n=2$

The following two lemmas give the explicit form of $\tilde{V}(\xi_{1},\xi_{2})$ and bounds on the heat kernel that imply the required bound.

Lemma 3.8**.**

[TABLE]

Proof.

By (3.35) and using that $V_{s}(\xi)=z_{s}=z_{0}e^{-\frac{\beta}{2}C_{s}(0,0)}$ by (3.36),

[TABLE]

Let $\sigma=\sigma_{1}\sigma_{2}$. By (3.28), $-\beta C_{s}(0,0)+W_{s}(\xi_{1},\xi_{2})=\sigma\beta C_{s}(x_{1},x_{2})$, so the integral can be evaluated as

[TABLE]

which after rearranging gives

[TABLE]

Lemma 3.9**.**

Let $U_{t}(x,y)=e^{\beta C_{t}(x,y)}-1$. The following bounds hold for $t\geqslant 0$, $f:\Lambda\to\mathbb{R}$, $\beta<8\pi$:

[TABLE]

and again analogous estimates hold in the conservative case.

Proof.

The lemma again follows from estimates for the heat kernel and is given in Appendix A. ∎

Proof of Proposition 3.6.

We first consider $V^{(2,\pm)}$. By (3.61) and (3.65),

[TABLE]

which is analogous to the bound for $\beta<4\pi$ and thus gives

[TABLE]

exactly as in (3.53). On the other hand, the neutral contribution to $V^{(2)}$ is given by

[TABLE]

Therefore

[TABLE]

By (3.66), the right-hand side is bounded by $O_{\beta}(|z_{t}|^{2}\ell_{t}^{4}\vartheta_{t}^{2})|f|_{2}^{2}=O_{\beta}(|\mathsf{z}_{t}|^{2}\vartheta_{t}^{2})|f|_{2}^{2}$. ∎

Remark 3.10**.**

Similarly as in (3.66), for $t>0$, $f:\Lambda\to\mathbb{R}$, $\beta<6\pi$, assuming (3.17), we have

[TABLE]

see Appendix A. Therefore, as in (3.70),

[TABLE]

provided that $\mathsf{z}_{t}=O(1)$. Exactly as in (3.68), the same bound holds for $V^{(2,\pm)}$, and as in (3.57) for $V-V^{(2)}$. In summary, whenever $|\mathsf{z}_{t}|$ is sufficiently small and (3.17) holds,

[TABLE]

3.7.  Proof of Proposition 3.7: the terms $n>2$

To bound the contributions due to (3.61), we need the following bounds on the heat kernel. For the statement of the bounds, we set

[TABLE]

Lemma 3.11**.**

Let $U_{t}(x,y)=e^{\beta C_{t}(x,y)}-1$. The following bounds hold for $t\geqslant 0$, $\beta<6\pi$:

[TABLE]

and the same bounds hold with the roles of the $x_{i}$ exchanged. Also, for all $t>s>0$, $x_{i}\in\Lambda$,

[TABLE]

Again analogous estimates hold in the conservative case.

Proof.

The lemma again follows from estimates for the heat kernel and is given in Appendix A. ∎

Lemma 3.12**.**

Let $\beta<6\pi$. Then $\|\tilde{V}_{t}^{(3)}\|\lesssim|z_{t}|^{3}\ell_{t}^{4}$. Analogous bounds hold in the conservative case.

Proof.

We start from (3.5). We assume $I_{1}=\{1,2\}$, $I_{2}=\{3\}$ since the other cases are analogous. We first consider the case that $\xi_{I_{1}}$ is neutral. Then

[TABLE]

By the definition of $W$ in (3.28) and by (3.78),

[TABLE]

By (3.76),

[TABLE]

Substituting these bounds into (3.7), this shows that the contribution to $\|\tilde{V}_{t}^{(3)}\|$ from neutral $\xi_{I_{1}}$ is bounded by

[TABLE]

where we used (3.23).

We turn now to the charged case $\sigma_{1}=\sigma_{2}$. Note that (3.80) follows as above if $\sigma_{3}=-\sigma_{1}$ and in fact holds with the better lower bound $\frac{3\beta}{4\pi}\log(\ell_{t}/\ell_{s})-O(1)$ by positive definiteness of $C_{t}-C_{s}$ if $\sigma_{3}=\sigma_{1}$, i.e., if all charges are the same. From the explicit form (3.61) of $\tilde{V}_{s}(\xi_{1},\xi_{2})$, we thus get

[TABLE]

As the sum over $x_{3}$ can be controlled uniformly in $x_{1},x_{2}$ by $O(\ell_{t}^{2}\vartheta_{t}^{2})$ thanks to (3.18) and then the sum over $x_{2}$ can be estimated by $O(\ell_{t}^{2})$ thanks to (3.65), we end up with the same upper bound as in (3.82). This completes the charged case. ∎

Lemma 3.13**.**

Let $\beta<6\pi$ and assume (3.17). Then $\|\tilde{V}_{t}^{(4)}\|\lesssim|z_{t}|^{4}\ell_{t}^{6}$. Analogous bounds hold in the conservative case.

Proof.

We again start from (3.5). Up to permutation of the indices, there are terms with $|I_{1}|=1$, $|I_{2}|=3$ and $|I_{1}|=|I_{2}|=2$. We begin with the case $|I_{1}|=1$ and $|I_{1}|=3$. Using that $|\mathsf{\dot{u}}_{s}|\lesssim\ell_{s}^{2}\vartheta_{s}^{2}$ and that $\|\tilde{V}_{s}^{(1)}\|\lesssim|z_{s}|$ and $\|\tilde{V}_{s}^{(3)}\|\lesssim|z_{s}|^{3}\ell_{s}^{4}$ (by (3.36) and Lemma 3.12),

[TABLE]

and we obtain the claimed bound exactly as in the proof for $\beta<4\pi$.

In the remainder of the proof we bound the terms with $|I_{1}|=|I_{2}|=2$. We begin with the case that $\xi_{I_{1}}$ and $\xi_{I_{2}}$ are both neutral. Up to permutation of the indices, we may then assume $\xi_{I_{1}}=((x_{1},+1),(x_{2},-1))$ and $\xi_{I_{2}}=((x_{3},+1),(x_{4},-1))$. By (3.61), using $\mathsf{\dot{u}}_{t}(\xi_{1},\xi_{j})+\mathsf{\dot{u}}_{t}(\xi_{2},\xi_{j})=\sigma_{1}\sigma_{j}(\mathsf{\dot{C}}_{t}(x_{1},x_{j})-\mathsf{\dot{C}}_{t}(x_{2},x_{j}))$ and analogously for the sum over $j$,

[TABLE]

Hence, by (3.77) and (3.22) for $\beta<6\pi$,

[TABLE]

In the case that $I_{1}$ is neutral and $I_{2}$ is charged, we similarly use

[TABLE]

By (3.76), the first bracket is bounded by

[TABLE]

Since $\xi_{I_{2}}$ is charged, the contribution from $V(\xi_{I_{2}})$ term is bounded using (3.65) by

[TABLE]

So altogether these contributions to (3.7) are again bounded using (3.22) (and $\beta<6\pi$) by

[TABLE]

Again the case that $\xi_{I_{1}}$ and $\xi_{I_{2}}$ are both charged is easier and analogous to the proof for $\beta<4\pi$ so omitted. ∎

Lemma 3.14**.**

Let $\beta<6\pi$ and assume (3.17). Then $\|\tilde{V}_{t}^{(n)}\|\leqslant n^{n-2}|z_{t}|^{n}(C_{\beta}\ell_{t}^{2})^{n-1}$ for all $n\geqslant 5$. Analogous bounds hold in the conservative case.

Proof.

Similarly as in the proof of (3.39), we make the inductive assumption that, for some $n\geqslant 4$, the bound (3.56) holds for all $1\leqslant k\leqslant n$, $k\neq 2$. By (3.36) and Lemmas 3.12-3.13, the inductive assumption is verified for $n=4$. To advance the induction we again start from

[TABLE]

For $|I_{1}|=n-k\neq 2$ and $|I_{2}|=k\neq 2$, we use

[TABLE]

and bound the terms on the right-hand side using the inductive assumption. Then exactly as in the proof for $\beta<4\pi$, i.e., of (3.39), the result is

[TABLE]

The terms with $|I_{1}|=2$ or $|I_{2}|=2$ require special treatment. By symmetry we may assume that $|I_{1}|=2$ and that $I_{1}=\{1,2\}$ and $I_{2}=\{3,\dots,n\}$ with $n\geqslant 5$. If $\xi_{I_{1}}$ is neutral, we use

[TABLE]

By (3.76), the first bracket is bounded by $O_{\beta}(z_{t}^{2}\ell_{t}^{4}\vartheta_{t}^{2})$, while for the second term involving $V(\xi_{I_{2}})$, using inductive assumption for $\tilde{V}(\xi_{I_{2}})$ (note that $n-2\geqslant 3$) to get

[TABLE]

So altogether these contributions to (3.7) are bounded by (using again (3.22) for $\beta<6\pi$),

[TABLE]

where in the last bound we have chosen $C_{\beta}$ sufficiently large (independently of $n$). Summing over the $\binom{n}{2}\leqslant n^{2}$ choices for $I_{1},I_{2}$ with $|I_{1}|=2$ leads to the expected upper bound. The charged case holds in the same way. ∎

Proof of Proposition 3.7.

The bounds (3.56) follows by combining the previous three lemmas. ∎

3.8.  Proofs of Propositions 3.1-3.2 without (3.5)

Finally, we remove the assumption (3.5) at the cost of constants that are uniform in $\varepsilon$ but not uniform in $L$. For $t\leqslant t_{0}$, where $t_{0}$ is sufficiently small but of order $1/\varepsilon^{2}$, we can apply the same analysis as before. On the other hand, for $t\geqslant t_{0}$, a very crude argument is sufficient to show that the Hessian of the effective potential is bounded from below uniformly in $\varepsilon$. Our starting point for this is (2.15), i.e.,

[TABLE]

The input from the previous analysis is summarised in the following lemma.

Lemma 3.15**.**

Let $\beta<6\pi$. Then there is a constant $\alpha=\alpha(\beta)>0$ such that for all $t\geqslant 0$ satisfying $|\mathsf{z}_{t}|\leqslant\alpha$ and (3.17), the following bounds hold uniformly in $\varphi\in X$, $f\in X$, and $x\in\Lambda$:

[TABLE]

Proof.

For $\beta<4\pi$, these bounds follow exactly as in (3.52)-(3.53). For $\beta<6\pi$, the bound on the Hessian is as in (3.55) and (3.57), and for $\nabla V_{t}$, see (3.73). ∎

Proof of Theorems 3.1-3.2 without (3.5).

From (3.18), recall that $e^{-\frac{\beta}{2}C_{t}(0,0)}\asymp\ell_{t}^{-\beta/4\pi}$ and hence that $|z_{t}|\asymp\varepsilon^{2}(\varepsilon\ell_{t})^{-\beta/4\pi}|z|$ and $|\mathsf{z}_{t}|\asymp(\varepsilon\ell_{t})^{2-\beta/4\pi}|z|$. Here $a\asymp b$ denotes that $c_{\beta}\leqslant a/b\leqslant 1/c_{\beta}$ for some constant $c_{\beta}>0$. Let $t_{\alpha}>0$ be such that $|\mathsf{z}_{t_{\alpha}}|=\alpha$. Thus $\varepsilon\ell_{t_{\alpha}}\asymp(\alpha/|z|)^{1/(2-\beta/4\pi)}$ and hence

[TABLE]

Also, with $t_{m,L}=\varepsilon^{-2}(m^{-2}\wedge L^{2})$ as in (3.17),

[TABLE]

We choose $t_{0}=t_{\alpha}\wedge t_{m,L}$ so that, since $|z_{t}|$ in decreasing in $t$ (see (3.19)),

[TABLE]

With this and since $|\Lambda|=\varepsilon^{-2}L^{2}$, it follows from (3.98) that, uniformly in $\varphi$,

[TABLE]

For any $t\geqslant t_{0}$, by the Cauchy-Schwarz inequality and $|Q_{t-t_{0}}f|_{2}\leqslant\vartheta_{t-t_{0}}|f|_{2}$, in particular,

[TABLE]

Similarly, by (3.97),

[TABLE]

Substituting (3.103)-(3.104) into (3.96), using that $\bm{P}_{t_{0},t}$ is a Markov operator, we conclude that, for all $t\geqslant t_{0}$,

[TABLE]

For $t\leqslant t_{0}$, we have $\dot{\mu}_{t}=O_{\beta}(|z_{t}|\vartheta_{t}^{2})=O_{\beta}(|z|)\varepsilon^{2}\vartheta_{t}^{2}$ exactly as in the proofs of the theorems in the case (3.5). In summary, for all $t\geqslant 0$,

[TABLE]

with $\mu^{*}(\beta,z,m,L)$ independent of $\varepsilon$. From this bound, the remainder of the proof is the same as in the case (3.5). ∎

Appendix A Heat kernel estimates: proof of Lemmas 3.3 and 3.9-3.11

In this appendix, we prove Lemmas 3.3 and 3.9-3.11. These follow from standard estimates for the lattice heat kernel $p_{t}(x)=e^{t\Delta}(0,x)$ on $\mathbb{Z}^{d}$ and its torus version $p_{t}^{L}(x)=\sum_{y\in\mathbb{Z}^{d}}p_{t}(x+Ly)$, where $L\in\mathbb{N}$. Throughout the appendix, $\Delta$ and $\nabla$ denote the lattice Laplacian and derivative on $\mathbb{Z}^{d}$, not the Laplacian and gradient on $\mathbb{R}^{\Lambda}$.

A.1.  Bounds on the heat kernel

We begin by collecting estimates on the heat kernel on $\mathbb{Z}^{d}$. To state these, let $\alpha$ be a sequence of $|\alpha|\equiv k$ unit vectors $\alpha_{1},\dots,\alpha_{k}$ in $\mathbb{Z}^{d}$, i.e., $\alpha_{i}\in\{e_{1\pm},\dots,e_{d\pm}\}$ is one of the $2d$ unit vectors $e_{i\pm}$ in $\mathbb{Z}^{d}$, and write $\nabla^{\alpha}=\prod_{i=1}^{k}\nabla_{\alpha_{i}}$ with $\nabla_{e}f(x)=f(x+e)-f(x)$ the lattice gradient. For $x\in\mathbb{Z}^{d}$, $|x|$ denotes any fixed norm unless stated.

Lemma A.1**.**

The heat kernel $p_{t}$ on $\mathbb{Z}^{d}$ satisfies the following upper bounds for $t\geqslant 1$, $x\in\mathbb{Z}^{d}$, and all sequences of unit vectors $\alpha$:

[TABLE]

as well as the following asymptotics if $d=2$, for $t\geqslant 1$ and $x\neq 0$,

[TABLE]

Moreover, the heat kernel $p_{t}^{L}$ on a discrete torus of side length $L$ satisfies, for $t\geqslant 1$, $|x|_{\infty}<L/2$,

[TABLE]

and the mean [math] heat kernel on the torus is given by $p_{t}^{0,L}(x)=p^{L}_{t}(x)-1/L^{2}$.

Proof.

Writing $\alpha_{i}=e_{j\sigma_{j}}$ with $j\in\{1,\dots,d\}$ and $\sigma_{j}\in\{\pm\}$ for each $i\in\{1,\dots,|\alpha|\}$, the bound (A.1) can be seen by writing $\nabla^{\alpha}p_{t}(x)$ in its Fourier representation:

[TABLE]

For $t\geqslant 1$, the integrand is analytic on a strip $k\in(\mathbb{R}+i[-c,c])^{d}$ with $c>0$ independent of $t$, and hence (A.1) decays exponentially in $|x|$ (see, e.g., [41, Chapter I.4, Exercise 4]). The first estimate in (A.2) is standard and straightforward to verify by writing the left-hand side in terms of the Fourier transform; we thus omit its proof. The second estimate in (A.2) is similarly standard if $t=\infty$ in which case the left-hand side is the Green function of the discrete Laplacian:

[TABLE]

This estimate can be found, for example, in [39, page 198] or [46, Theorem 4.4.4] (with normalisation there differing by a factor $2d=4$). To prove the second estimate in (A.2) for $0<|x|\leqslant\sqrt{t}$, we use that by (A.1) with $|\alpha|=1$,

[TABLE]

which using (A.5) implies

[TABLE]

For $|x|\geqslant\sqrt{t}$, we use that the first bound in (A.2) (and $p_{t}(0)\leqslant 1$ for $t<1$) implies

[TABLE]

and hence with (A.1) to bound $p_{s}(x)$,

[TABLE]

where the integral is bounded by a multiple of

[TABLE]

This completes the proof of (A.2).

For the torus of side length $L$, we use that $p_{t}^{L}(x)=\sum_{y\in\mathbb{Z}^{d}}p_{t}(x+Ly)$ and set $|x|_{L}=\inf_{y\in\mathbb{Z}^{d}}|x+Ly|$. Then

[TABLE]

since the remainder between the left-hand side and the first term on the right-hand side of the last equation can be controlled by (approximating the sum by an integral and using polar coordinates)

[TABLE]

This shows the estimates (A.3).

The expression for the mean [math] heat kernel follows from $p_{t}^{0,L}(x)=(\delta_{0},Pe^{\Delta t}P\delta_{x})=(\delta_{0}-1/L^{2},e^{\Delta t}(\delta_{x}-1/L^{2}))=p_{t}^{L}(x)-2/L^{2}+1/L^{2}=p_{t}^{L}(x)-1/L^{2}$ with the projection $P$ from (3.7). ∎

A.2.  Proof of Lemma 3.3

We recall the definition $\dot{C}_{t}(x)=p_{t}^{L_{\varepsilon}}(x)e^{-\varepsilon^{2}m^{2}t}=p_{t}^{L_{\varepsilon}}(x)\vartheta_{t}^{2}$. Lemma 3.3 is an elementary combination of the estimates from Lemma A.1, whose details are given as follows.

Proof of Lemma 3.3.

Applying (A.1) and (A.3) with $x=0$ to the torus of side length $L_{\varepsilon}=L/\varepsilon$ and, for $t\geqslant 1$, we have

[TABLE]

By the assumption (3.17), either $t\leqslant 1/\varepsilon^{2}m^{2}$ or $Lm\geqslant 1$ holds. By the above bound, if $Lm\geqslant 1$, the contribution to $C_{t}(0)$ from $t\geqslant 1/\varepsilon^{2}m^{2}$ is negligible since

[TABLE]

For $t\leqslant L^{2}/\varepsilon^{2}$ (and thus for $t\leqslant 1/m^{2}\varepsilon^{2}$ when $Lm\geqslant 1$), we may moreover replace $p_{t}^{L_{\varepsilon}}$ by $p_{t}$ since

[TABLE]

Finally, the contribution to $\dot{C}_{t}(0)$ from the infinite volume heat kernel $p_{t}(0)$ is

[TABLE]

which integrated up to $t\leqslant 1/\varepsilon^{2}m^{2}$ gives the main contribution

[TABLE]

This shows the first estimate in (3.18). The second estimate is straightforward since $\mathsf{\dot{C}}_{s}(x,y)=\mathsf{\dot{C}}_{s}(0,x-y)\geqslant 0$ and the fact that the heat kernel defines a probability density immediately imply

[TABLE]

Finally, in the conservative case the estimates are unchanged since

[TABLE]

and

[TABLE]

A.3.  Proof of Lemmas 3.9-3.11

To prepare for the proofs of the lemmas, we state the following consequences of Lemma A.1 in the notation used in the lemmas. In particular, recall (3.74)-(3.75). For $x\in\Lambda$, abusing notation slightly, we write $|x|$ for the torus distance $|x|_{L_{\varepsilon}}=\inf_{y\in\mathbb{Z}^{d}}|x+L_{\varepsilon}y|$. In particular, $|x|=O(L_{\varepsilon})$ for all $x\in\Lambda$. Moreover, in all of the following lemmas, we impose the assumption (3.17) without stating it explicitly.

Lemma A.2**.**

The following estimates hold for $\mathsf{\dot{C}}_{t}$, $C_{t}$ for $t\geqslant 1$ and $|x-y|\geqslant 1$:

[TABLE]

The first bounds also implies that

[TABLE]

For any $c^{\prime}>0$ small enough,

[TABLE]

The same estimates hold with $\mathsf{\dot{C}}_{t}$ replaced by $\ell_{t}\vartheta_{t}\mathsf{Q}_{t}$, and if $\mathsf{\dot{C}}_{t}$ and $\mathsf{Q}_{t}$ are replaced by $\mathsf{\dot{C}}_{t}^{0}$ and $\mathsf{Q}_{t}^{0}$.

Proof.

The estimates (A.21) follow easily from those for the heat kernel in (A.1)-(A.3). Indeed, the second bound in (A.21) is a special case of (A.1) and (A.3):

[TABLE]

where in the last inequality we used that $\ell_{t}/L_{\varepsilon}\leqslant 1$ follows from (3.17) and the definition of $\ell_{t}$ in (3.15). Indeed, by (3.17), either $t\leqslant L_{\varepsilon}^{2}$ which implies $\ell_{t}\leqslant L_{\varepsilon}$, or otherwise $Lm\geqslant 1$ and then also $\ell_{t}/L_{\varepsilon}=(\sqrt{t}\wedge 1/(\varepsilon m))/(L/\varepsilon)\leqslant\sqrt{\varepsilon^{2}m^{2}t}\wedge 1\leqslant 1$.

For the first bound in (A.21) we note that (A.2) implies

[TABLE]

The additional factor $e^{-\varepsilon^{2}m^{2}s}$ multiplying $p_{s}(x)$ leads to the replacement of $\sqrt{t}$ by $\ell_{t}$ exactly as in the proof of (3.18). By an analogous calculation, the same formula holds with the discrete heat kernel replaced by the continuous one, i.e.,

[TABLE]

from which (A.22) follows after taking into account the additional factor $e^{-\varepsilon^{2}m^{2}s}$ as before.

To verify (A.23)-(A.24), for $x,y\in\mathbb{Z}^{d}$, let $\gamma_{xy}$ be a path from $x$ to $y$ of length $|x-y|$ where $|x|$ denotes the $1$-norm in this proof. Then (A.1) and (A.3) imply

[TABLE]

For $u\in\gamma_{xy}$, we have $|x-z|\leqslant|x-u|+|u-z|\leqslant|x-y|+|u-z|$, and we deduce from the symmetric estimate in $y$ that $-|u-z|\leqslant-|x-y|-|x-z|/2-|y-z|/2$. Choosing $c^{\prime}<c$, we get

[TABLE]

This completes (A.23). Analogously, again applying (A.1) and (A.3) and choosing $c^{\prime}<c$, we get

[TABLE]

using that $|x-w|\leqslant|x-u|+|u-v|+|v-w|\leqslant|x-y|+|u-v|+|w-z|$. ∎

Lemma A.3**.**

For all $x,y,z\in\Lambda$, $0\leqslant s\leqslant t$,

[TABLE]

Proof.

It suffices to assume that $s\geqslant 1$. Throughout this proof, $|x|$ denotes the Euclidean norm. Suppose first that $|x-y|\leqslant|x-z|\wedge|y-z|$. We will show that

[TABLE]

Indeed, this bound follows from the following two estimates: using (A.1) with $|\alpha|=0$ for the first bound and with $|\alpha|=1$ for the second bound, and also (A.3) for the error due to periodicity,

[TABLE]

Here we have used that the remainder in (A.3) due to the periodicity is bounded by

[TABLE]

when $Lm\geqslant 1$, and that an analogous bound holds when instead $t\leqslant\varepsilon^{-2}(m^{-2}\wedge L^{2})$. The bound (A.31) then follows from (A.32) and $(C_{t}-C_{s})(0,0)-(C_{t}-C_{s})(x,y)\geqslant 0$ which holds by the positive definiteness of $C_{t}-C_{s}$ and translation invariance.

The same argument as above also applies if $|y-z|\leqslant|x-z|\wedge|x-y|$. Therefore suppose that $|x-z|\leqslant|x-y|\wedge|y-z|$. From (A.22) recall that

[TABLE]

Since $e^{-|x-z|^{2}/2u}\geqslant e^{-|y-z|^{2}/2u}$ therefore

[TABLE]

The conclusion (A.31) now again follows from $(C_{t}-C_{s})(0,0)-(C_{t}-C_{s})(x,y)\geqslant 0$. ∎

Lemma A.4**.**

Let $U_{t}(x)=e^{\beta C_{t}(0,x)}-1$. Then for $\beta<2\pi(k+2)$ and sufficiently small $c^{\prime}>0$,

[TABLE]

The analogous estimate holds in the conservative case.

Proof.

By (A.21), $C_{s}(0,x)=-\frac{1}{2\pi}\log(|x|/\ell_{s}\wedge 1)+O(1)$ and $|\mathsf{\dot{C}}_{s}(0,x)|\lesssim\vartheta_{s}^{2}e^{-c|x|/\sqrt{s}}$. Therefore

[TABLE]

Choosing $c^{\prime}<c/2$, we get $e^{c^{\prime}|x|/\sqrt{t}}e^{-c|x|/\sqrt{s}}\leqslant e^{-\frac{1}{2}c|x|/\sqrt{s}}$ for $t\geqslant s$. Furthermore

[TABLE]

holds if $2+k>\beta/2\pi$ and $s\geqslant 1$. Therefore

[TABLE]

The bounds are the same in the conservative case. ∎

With the above preparation, we now prove Lemmas 3.9-3.11.

Proof of (3.65).

For (3.65), we use $C_{t}(0,x)\geqslant 0$ which with $1-e^{-x}\leqslant x$ for $x\geqslant 0$ gives the claim

[TABLE]

In the conservative case, $C_{t}^{0}(x)\geqslant-1/L^{2}$ and the claim follows similarly from $|1-e^{-x}|\leqslant 2|x|$ for $x\geqslant-1$. ∎

Proof of (3.66).

For sufficiently small $c^{\prime}>0$, we write

[TABLE]

where

[TABLE]

By (A.38), then $\sup_{x}\sum_{y}A_{xy}\lesssim\ell_{t}^{2}$ for $c^{\prime}>0$ small enough. By (A.23) for $\ell_{t}\vartheta_{t}\mathsf{Q}_{t}$ instead of $\mathsf{\dot{C}}_{t}$ and the inequality $2ab\leqslant a^{2}+b^{2}$, we have for $x\neq y$,

[TABLE]

Thus there are positive $M_{xy}=M_{yx}=O(\vartheta_{t}\ell_{t}^{-1}e^{-c^{\prime}|x-y|/\ell_{t}})$, i.e., $\sup_{x}\sum_{y}M_{xy}\lesssim\ell_{t}\vartheta_{t}$, such that

[TABLE]

Then (using $(a+b)^{2}\leqslant 2a^{2}+2b^{2}$ and $A_{xy}=A_{yx}$),

[TABLE]

Similarly (with $2|ab|\leqslant a^{2}+b^{2}$ and $M_{xy}=M_{yx}$)

[TABLE]

Therefore

[TABLE]

Since $\sup_{x}\sum_{y}A_{xy}\lesssim\ell_{t}^{2}$ and $\sup_{x}\sum_{y}M_{xy}\lesssim\vartheta_{t}\ell_{t}$, the desired bound $\lesssim\vartheta_{t}^{2}\ell_{t}^{4}$ follows. The bounds are unchanged in the conservative case. ∎

Proof of (3.71).

We proceed analogously to the proof of (3.66), i.e., for sufficiently small $c^{\prime}>0$, we write

[TABLE]

where

[TABLE]

By (A.38), again $\sup_{x}\sum_{y}A_{xy}\lesssim\ell_{t}^{2}$ for $c^{\prime}>0$ small enough, but now using that $\beta<6\pi$ due to the different power in the definition of $A_{xy}$. The bound for $B_{xy}$ is the same. From this, we conclude

[TABLE]

Since $\mathsf{Q}_{t}=\ell_{t}Q_{t}$, this is (3.71). The bounds are unchanged in the conservative case. ∎

Proof of (3.76).

By (A.23) and (A.38) (with $\beta<6\pi$), one can find $c^{\prime}>0$ small enough such that

[TABLE]

where a factor $\ell_{t}^{2}$ comes first by summing over $x_{3}$ and another factor $\ell_{t}^{2}$ from (A.38). The same applies when the roles of $x_{1},x_{2},x_{3}$ in the $\sup$ and sum are exchanged. The bounds are unchanged in the conservative case. ∎

Proof of (3.77).

By (A.24), there is $c^{\prime}>0$ small enough such that

[TABLE]

and using (A.38) both for the sum over $x_{2}$ and $x_{4}$ (with $\beta<6\pi$), as well as the elementary bound $\sup_{x_{1}}\sum_{x_{3}}e^{-c|x_{1}-x_{3}|/\ell_{t}}\lesssim\ell_{t}^{2}$, this implies

[TABLE]

with one factor $\ell_{t}^{2}$ from each of the sums. The bounds are unchanged in the conservative case. ∎

Acknowledgements

We thank David Brydges for invaluable discussions and for feedback on preliminary versions of this manuscript. We also thank Felix Otto for pointing out the resemblence of our construction with some works related to the Ricci flow. We acknowledge the support of ANR-15-CE40-0020-01 grant LSD. We would also like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Scaling limits, rough paths, quantum field theory” when work on this paper was undertaken; this work was supported by EPSRC Grant Number EP/R014604/1.