Phase Transitions for $\phi ^4_3$

Abstract

We establish a surface order large deviation estimate for the magnetisation of low temperature $\phi ^4_3$. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the $\phi ^4_3$ singular stochastic PDE. Our main technical contributions are contour bounds for $\phi ^4_3$, which extends 2D results by Glimm et al. (Commun Math Phys 45(3):203–216, 1975). We adapt an argument by Bodineau et al. (J Math Phys 41(3):1033–1098, 2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of $\phi ^4_3$ whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for $\phi ^4_3$ developed recently by Barashkov and Gubinelli (Duke Math. J. 169(17):3339–3415, 2020).

1 Introduction

We study the behaviour of the average magnetisation

$$\begin{aligned} {\mathfrak {m}}_N(\phi ) = \frac{1}{N^3} \int _{{\mathbb {T}}_N}\phi (x) dx \end{aligned}$$

for fields $\phi$ distributed according to the measure ${\nu _{\beta ,N}}$ with formal density

$$\begin{aligned} d{\nu _{\beta ,N}}(\phi ) \propto \exp \Big ( - \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (\phi (x)) + \frac{1}{2} |\nabla \phi (x)|^2 dx \Big )\prod _{x\in {\mathbb {T}}_N}d\phi (x) \end{aligned}$$

(1.1)

in the infinite volume limit $N \rightarrow \infty$. Above, ${\mathbb {T}}_N= ({\mathbb {R}}/N{\mathbb {Z}})^3$ is the 3D torus of sidelength $N \in {\mathbb {N}}$, $\prod _{x\in {\mathbb {T}}_N}d\phi (x)$ is the (non-existent) Lebesgue measure on fields $\phi :{\mathbb {T}}_N\rightarrow {\mathbb {R}}$, $\beta > 0$ is the inverse temperature, and ${\mathcal {V}}_\beta : {\mathbb {R}}\rightarrow {\mathbb {R}}$ is the symmetric double-well potential given by ${\mathcal {V}}_\beta (a) = \frac{1}{\beta }(a^2-\beta )^2$ for $a \in {\mathbb {R}}$.

${\nu _{\beta ,N}}$ is a finite volume approximation of a $\phi ^4_3$ Euclidean quantum field theory [Gli68, GJ73, FO76]. Its construction, first in finite volumes and later in infinite volume, was a major achievement of the constructive field theory programme in the ’60s-’70s: Glimm and Jaffe made the first breakthrough in [GJ73] and many results followed [Fel74, MS77, BCG+80, BFS83, BDH95, MW17, GH18, BG19]. The model in 2D was constructed earlier by Nelson [Nel66]. In higher dimensions there are triviality results: in dimensions $\geqslant 5$ these are due to Aizenman and Fröhlich [Aiz82, Frö82], whereas the 4D case was only recently done by Aizenman and Duminil-Copin [ADC20]. By now it is also well-known that the $\phi ^4_3$ model has significance in statistical mechanics since it arises as a continuum limit of Ising-type models near criticality [SG73, CMP95, HI18].

It is natural to define ${\nu _{\beta ,N}}$ using a density with respect to the centred Gaussian measure $\mu _N$ with covariance $(-\Delta )^{-1}$, where $\Delta$ is the Laplacian on ${\mathbb {T}}_N$ (see Remark 1.1 for how we deal with the issue of constant fields/the zeroeth Fourier mode). However, in 2D and higher $\mu _N$ is not supported on a space of functions and samples need to be interpreted as Schwartz distributions. This is a serious problem because there is no canonical interpretation of products of distributions, meaning that the nonlinearity $\int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (\phi (x)) dx$ is not well-defined on the support of $\mu _N$. If one introduces an ultraviolet (small-scale) cutoff $K>0$ on the field to regularise it, then one sees that the nonlinearities ${\mathcal {V}}_\beta (\phi _K)$ fail to converge as the cutoff is removed—there are divergences. The strength of these divergences grow as the dimension grows: they are only logarithmic in the cutoff in 2D, whereas they are polynomial in the cutoff in 3D. In addition, ${\nu _{\beta ,N}}$ and $\mu _N$ are mutually singular [BG20] in 3D, which produces technical difficulties that are not present in 2D.

Renormalisation is required in order to kill these divergences. This is done by looking at the cutoff measures and subtracting the corresponding counter-term $\int _{{\mathbb {T}}_N}\delta m^2(K) \phi ^2_K$ where $\phi _K$ is the field cutoff at spatial scales less than $\frac{1}{K}$ and the renormalisation constant $\delta m^2(K) = \frac{C_1}{\beta } K - \frac{C_2}{\beta ^2} \log K$ for specific constants $C_1, C_2 > 0$ (see Sect. 2). If these constants are appropriately chosen (i.e. by perturbation theory), then a non-Gaussian limiting measure is obtained as $K \rightarrow \infty$. This construction yields a one-parameter family of measures ${\nu _{\beta ,N}}={\nu _{\beta ,N}}(\delta m^2)$ corresponding to bounded shifts of $\delta m^2(K)$.

Remark 1.1

For technical reasons, we work with a massive Gaussian free field as our reference measure. We do this by introducing a mass $\eta > 0$ into the covariance. This resolves the issue of the constant fields/zeroeth Fourier mode degeneracy. In order to stay consistent with (1.1), we subtract $\int _{{\mathbb {T}}_N}\frac{\eta }{2} \phi ^2 dx$ from ${\mathcal {V}}_\beta (\phi )$.

Once we have chosen $\eta$, it is convenient to fix $\delta m^2$ by writing the renormalisation constants in terms of expectations with respect to $\mu _N(\eta )$. The particular choice of $\eta$ is inessential since one can show that changing $\eta$ corresponds to a bounded shift of $\delta m^2$ that is $O\Big (\frac{1}{\beta }\Big )$ as $\beta \rightarrow \infty$.

The large-scale behaviour of ${\nu _{\beta ,N}}$ depends heavily on $\beta$ as $N \rightarrow \infty$. To see why, note that $a \mapsto {\mathcal {V}}_\beta (a)$ has minima at $a = \pm {\sqrt{\beta }}$ with a potential barrier at $a=0$ of height $\beta$, so the minima become widely separated by a steep barrier as $\beta \rightarrow \infty$. Consequently, ${\nu _{\beta ,N}}$ resembles an Ising model on ${\mathbb {T}}_N$ with spins at $\pm {\sqrt{\beta }}$ (i.e. at inverse temperature $\beta > 0$) for large $\beta$. Glimm et al. [GJS75] exploited this similarity and proved phase transition for $\nu _\beta$, the infinite volume analogue of ${\nu _{\beta ,N}}$, in 2D using a sophisticated modification of the classical Peierls’ argument for the low temperature Ising model [Pei36, Gri64, Dob65]. See also [GJS76a, GJS76b]. Their proof relies on contour bounds for ${\nu _{\beta ,N}}$ in 2D that hold in the limit $N \rightarrow \infty$. Their techniques fail in the significantly harder case of 3D. However, phase transition for $\nu _\beta$ in 3D was established by Fröhlich, Simon, and Spencer [FSS76] using a different argument based heavily on reflection positivity. Whilst this argument is more general (it applies, for example, to some models with continuous symmetry), it is less quantitative than the Peierls’ theory of [GJS75]. Specifically, it is not clear how to use it to control large deviations of the (finite volume) average magnetisation ${\mathfrak {m}}_N$.

Although phase coexistence for $\nu _\beta$ has been established, little is known of this regime in comparison to the low temperature Ising model. In the latter model, the study of phase segregation at low temperatures in large but finite volumes was initiated by Minlos and Sinai [MS67, MS68], culminating in the famous Wulff constructions: due to Dobrushin, Kotecký, and Shlosman in 2D [DKS89, DKS92], with simplifications due to Pfister [Pfi91] and results up to the critical point by Ioffe and Schonmann [IS98]; and Bodineau [Bod99] in 3D, see also results up to the critical point by Cerf and Pisztora [CP00] and the bibliographical review in [BIV00, Section 1.3.4]. We are interested in a weaker form of phase segregation: surface order large deviation estimates for the average magnetisation ${\mathfrak {m}}_N$. For the Ising model, this was first established in 2D by Schonmann [Sch87] and later extended up to the critical point by Chayes, Chayes, and Schonmann [CCS87]; in 3D this was first established by Pisztora [Pis96]. These results should be contrasted with the volume order large deviations established for ${\mathfrak {m}}_N$ in the high temperature regime where there is no phase coexistence [CF86, Ell85, FO88, Oll88].

Our main result is a surface order upper bound on large deviations for the average magnetisation under ${\nu _{\beta ,N}}$.

Theorem 1.2

Let $\eta > 0$ and ${\nu _{\beta ,N}}= {\nu _{\beta ,N}}(\eta )$ as in Remark 1.1. For any $\zeta \in (0,1)$, there exists $\beta _0 = \beta _0(\zeta ,\eta ) > 0$, $C=C(\zeta , \eta )>0$, and $N_0 = N_0(\zeta ) \geqslant 4$ such that the following estimate holds: for any $\beta > \beta _0$ and any $N > N_0$ dyadic,

$$\begin{aligned} \frac{1}{N^2} \log \nu _{\beta ,N} \Big ( {\mathfrak {m}}_N \in (-\zeta {\sqrt{\beta }}, \zeta {\sqrt{\beta }}) \Big ) \leqslant - C {\sqrt{\beta }}. \end{aligned}$$

(1.2)

Proof

See Sect. 3.5. $\quad \square$

The condition that N is a sufficiently large dyadic in Theorem 1.2 comes from Proposition 3.8 (we also need that N is divisible by 4 to apply the chessboard estimates of Proposition 6.5). Our analysis can be simplified to prove Theorem 1.2 in 2D with $N^2$ replaced by N in (1.2).

Our main technical contributions are contour bounds for ${\nu _{\beta ,N}}$. As a result, the Peierls’ argument of [GJS75] is extended to 3D, thereby giving a second proof of phase transition for $\phi ^4_3$. The main difficulty is to handle the ultraviolet divergences of ${\nu _{\beta ,N}}$ whilst preserving the structure of the low temperature potential. We do this by building on the variational approach to showing ultraviolet stability for $\phi ^4_3$ recently developed by Barashkov and Gubinelli [BG19]. Our insight is to separate scales within the corresponding stochastic control problem through a coarse-graining into an effective Hamiltonian and remainder. The effective Hamiltonian captures the macroscopic description of the system and is treated using techniques adapted from [GJS76b]. The remainder contains the ultraviolet divergences and these are killed using the renormalisation techniques of [BG19].

Our next contribution is to adapt arguments used by Bodineau, Velenik, and Ioffe [BIV00], in the context of equilibrium crystal shapes of discrete spin models, to study phase segregation for $\phi ^4_3$. In particular, we adapt them to handle a block-averaged model with unbounded spins. Technically, this requires control over large fields.

1.1 Application to the dynamical $\phi ^4_3$ model

The Glauber dynamics of ${\nu _{\beta ,N}}$ is given by the singular stochastic PDE

$$\begin{aligned} \begin{aligned} (\partial _t - \Delta + \eta )\Phi&= -\frac{4}{\beta }\Phi ^3 + (4+\eta +\infty )\Phi + \sqrt{2}\xi \\ \Phi (0,\cdot )&= \phi _0 \end{aligned} \end{aligned}$$

(1.3)

where $\Phi \in S'({\mathbb {R}}_+\times {\mathbb {T}}_N)$ is a space-time Schwartz distribution, $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2} -\kappa }({\mathbb {T}}_N)$, the infinite constant indicates renormalisation (see Remark 6.16), and $\xi$ is space-time white noise on ${\mathbb {T}}_N$. The well-posedness of this equation, known as the dynamical $\phi ^4_3$ model, has been a major breakthrough in stochastic analysis in recent years [Hai14, Hai16, GIP15, CC18, Kup16, MW17, GH19, MW18].

In finite volumes the solution is a Markov process and its associated semigroup $({\mathcal {P}}_t^{\beta ,N})_{t \geqslant 0}$ is reversible and exponentially ergodic with respect to its unique invariant measure ${\nu _{\beta ,N}}$ [HM18a, HS19, ZZ18a]. As a consequence, there exists a spectral gap $\lambda _{\beta ,N}>0$ given by the optimal constant in the inequality:

$$\begin{aligned} \Big \langle \Big ({\mathcal {P}}_t^{\beta ,N} F \Big )^2 \rangle _{\beta ,N} - \Big ( \Big \langle {\mathcal {P}}_t^{\beta ,N} F \Big \rangle _{\beta ,N} \Big )^2 \leqslant e^{-\lambda _{\beta ,N}t} \Big (\langle F^2 \rangle _{\beta ,N} - \langle F \rangle _{\beta ,N}^2 \Big ) \end{aligned}$$

for suitable $F \in L^2({\nu _{\beta ,N}})$. $\lambda _{\beta ,N}^{-1}$ is called the relaxation time and measures the rate of convergence of variances to equilibrium. An implication of Theorem 1.2 is the exponential explosion of relaxation times in the infinite volume limit provided $\beta$ is sufficiently large.

Corollary 1.3

Let $\eta > 0$ and ${\nu _{\beta ,N}}= {\nu _{\beta ,N}}(\eta )$ as in Remark 1.1. Then, there exists $\beta _0=\beta _0(\eta )>0$, $C=C(\beta _0,\eta )$, and $N_0 \geqslant 4$ such that, for any $\beta > \beta _0$ and $N > N_0$ dyadic,

$$\begin{aligned} \frac{1}{N^2} \log \lambda _{\beta ,N} \leqslant - C{\sqrt{\beta }}. \end{aligned}$$

(1.4)

Proof

See Sect. 7. $\quad \square$

Corollary 1.3 is the first step towards establishing phase transition for the relaxation times of the Glauber dynamics of $\phi ^4$ in 2D and 3D. This phenomenon has been well-studied for the Glauber dynamics of the 2D Ising model, where a relatively complete picture has been established (in higher dimensions it is less complete). The relaxation times for the Ising dynamics on the 2D torus of sidelength N undergo the following trichotomy as $N \rightarrow \infty$: in the high temperature regime, they are uniformly bounded in N [AH87, MO94]; in the low temperature regime, they are exponential in N [Sch87, CCS87, Tho89, MO94, CGMS96]; at criticality, they are polynomial in N [Hol91, LS12]. It would be interesting to see whether the relaxation times for the dynamical $\phi ^4$ model undergo such a trichotomy.

1.2 Paper organisation

In Sect. 2 we introduce the renormalised, ultraviolet cutoff measures $\nu _{\beta ,N,K}$ that converge weakly to ${\nu _{\beta ,N}}$ as the cutoff is removed. In Sect. 3 we carry out the statistical mechanics part of the proof of Theorem 1.2. In particular, conditional on the moment bounds in Proposition 3.6, we develop contour bounds for ${\nu _{\beta ,N}}$. These contour bounds allow us to adapt techniques in [BIV00], which were developed in the context of discrete spin systems, to deal with ${\nu _{\beta ,N}}$.

In Sect. 4 we lay the foundation to proving Proposition 3.6 by introducing the Boué–Dupuis formalism for analysing the free energy of ${\nu _{\beta ,N}}$ as in [BG19]. We then use a low temperature expansion and coarse-graining argument within the Boué–Dupuis formalism in Sect. 5 to establish Proposition 5.1 which contains the key analytic input to proving Proposition 3.6.

In Sect. 6, we use the chessboard estimates of Proposition 6.5 to upgrade the bounds of Proposition 5.1 to those of Proposition 3.6. Chessboard estimates follow from the well-known fact that ${\nu _{\beta ,N}}$ is reflection positive. We give an independent proof of this fact by using stability results for the dynamics (1.3) to show that lattice and Fourier regularisations of ${\nu _{\beta ,N}}$ converge to the same limit. Then, in Sect. 7, we prove Corollary 1.3 showing that the spectral gaps for the dynamics decay in the infinite volume limit provided $\beta$ is sufficiently large.

We collect basic notations and analytic tools that we use throughout the paper in “Appendix A”.

2 The Model

In the following, we use notation and standard tools introduced in “Appendix A”.

Let $\eta > 0$. Denote by $\mu _N = \mu _N(\eta )$ the centred Gaussian measure with covariance $(-\Delta + \eta )^{-1}$ and expectation ${\mathbb {E}}_N$. Above, $\Delta$ is the Laplacian on ${\mathbb {T}}_N$. As pointed out in Remark 1.1, the choice of $\eta$ is inessential. We consider it fixed unless stated otherwise and we do not make $\eta$-dependence explicit in the notation.

Fix $\beta > 0$. Let ${\mathcal {V}}_\beta :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be given by

$$\begin{aligned} {\mathcal {V}}_\beta (a) = \frac{1}{\beta }(a^2 - \beta )^2 = \frac{1}{\beta }a^4 - 2a^2 + \beta . \end{aligned}$$

${\mathcal {V}}_\beta$ is a symmetric double well potential with minima at $a = \pm {\sqrt{\beta }}$ and a potential barrier at $a=0$ of height $\beta$.

Fix $\rho \in C^\infty _c({\mathbb {R}}^3;[0,1])$ rotationally symmetric; decreasing; and satisfying $\rho (x)=1$ for $|x| \in [0,c_\rho )$, where $c_\rho >0$. See Lemma 4.6 for why the last condition is important. Note that many of our estimates rely on the choice of $\rho$, but we omit explicit reference to this.

For every $K>0$, let $\rho _K$ be the Fourier multiplier on ${\mathbb {T}}_N$ with symbol $\rho _K(\cdot ) = \rho (\frac{\cdot }{K})$. For $\phi \sim \mu _N$, we denote $\phi _K = \rho _K \phi$. Note that $\phi _K$ is smooth. Let

(2.1)

where $\langle \cdot \rangle = \sqrt{\eta + 4\pi ^2|\cdot |}$. Note that as $K \rightarrow \infty$. The first four Wick powers of $\phi _K$ are given by the generalised Hermite polynomials:

We define the Wick renormalised potential by linearity:

$$\begin{aligned} :{\mathcal {V}}_\beta (\phi _K): = \frac{1}{\beta } : \phi _K^4: - 2 :\phi _K^2: + \beta . \end{aligned}$$

Let $\nu _{\beta ,N,K}$ be the probability measure with density

$$\begin{aligned} d\nu _{\beta ,N,K}(\phi ) = \frac{e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K)}}{{\mathscr {Z}}_{\beta ,N,K}} d\mu _N(\phi ). \end{aligned}$$

(2.2)

Above, ${\mathcal {H}}_{\beta ,N,K}$ is the renormalised Hamiltonian

$$\begin{aligned} {\mathcal {H}}_{\beta ,N,K}(\phi _K)&= \int _{{\mathbb {T}}_N}:{\mathcal {V}}_\beta (\phi _K): - \frac{\gamma _K}{\beta ^2} :\phi _K^2: - \delta _K - \frac{\eta }{2} :\phi _K^2: dx \end{aligned}$$

(2.3)

where $\gamma _K$ and $\delta _K$ are additional renormalisation constants given by (5.25) and (5.26), respectively, and ${\mathscr {Z}}_{\beta ,N,K} = {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K)}$ is the partition function.

Proposition 2.1

For every $\beta > 0$ and $N \in {\mathbb {N}}$, the measures $\nu _{\beta ,N,K}$ converge weakly to a non-Gaussian measure ${\nu _{\beta ,N}}$ on $S'({\mathbb {T}}_N)$ as $K \rightarrow \infty$. In addition, ${\mathscr {Z}}_{\beta ,N,K} \rightarrow {\mathscr {Z}}_{\beta ,N}$ as $K \rightarrow \infty$ and satisfies the following estimate: there exists $C=C(\beta ,\eta ) > 0$ such that

$$\begin{aligned} -CN^3 \leqslant -\log {\mathscr {Z}}_{\beta ,N} \leqslant CN^3. \end{aligned}$$

Proof

Proposition 2.1 is a variant of the classical ultraviolet stability for $\phi ^4_3$ first established in [GJ73]. Our precise formulation, i.e. the choice of $\gamma _\bullet$ and $\delta _\bullet$, is taken from [BG19, Theorem 1]. $\quad \square$

We write $\langle \cdot \rangle _{\beta ,N}$ and $\langle \cdot \rangle _{\beta ,N,K}$ for expectations with respect to ${\nu _{\beta ,N}}$ and $\nu _{\beta ,N,K}$, respectively.

Remark 2.2

The constants are, respectively, Wick renormalisation, (second order) mass renormalisation, and energy renormalisation constants. They all depend on $\eta$ and N. $\delta _K$ additionally depends on $\beta$ and is needed for the convergence of ${\mathscr {Z}}_{\beta ,N,K}$ as $K \rightarrow \infty$, but drops out of the definition of the cutoff measures (2.2).

Remark 2.3

In 2D a scaling argument [GJS76c] allows one to work with the measure with density proportional to

$$\begin{aligned} \exp \Big ( - \int _{{\mathbb {T}}_N}:{\mathcal {V}}_\beta (\phi _K): dx \Big ) d{\tilde{\mu }}_N(\phi ) \end{aligned}$$

where ${\tilde{\mu }}_N$ is the Gaussian measure with covariance $(-\Delta + {\sqrt{\beta }}^{-1})^{-1}$, i.e. a $\beta$-dependent mass. This measure is significantly easier to work with due to the degenerate mass when $\beta$ is large. In particular, it is easier to obtain contour bounds which, although suboptimal from the point of view of $\beta$-dependence, are sufficient for the Peierls’ argument in [GJS75] and for the analogue of our argument in Sect. 3 carried out in 2D. In 3D one cannot work with such a measure.

3 Surface Order Large Deviation Estimate

In this section we carry out the statistical mechanics part of the proof of Theorem 1.2. Recall that for large $\beta$, the the minima of potential ${\mathcal {V}}_\beta$ at $\pm {\sqrt{\beta }}$ are widely separated by a steep potential barrier of height $\beta$, so formally ${\nu _{\beta ,N}}$ resembles an Ising model at inverse temperature $\beta$. We use this intuition to prove contour bounds for ${\nu _{\beta ,N}}$ (see Proposition 3.2) conditional on certain moment bounds (see Proposition 3.6). The contour bounds are then used to adapt arguments from [BIV00] to prove Theorem 1.2.

3.1 Block averaging

Let $e_1, e_2, e_3$ be the standard basis for ${\mathbb {R}}^3$. We identify ${\mathbb {T}}_N$ with the set

$$\begin{aligned} \big \{ a_1 e_1 + a_2 e_2 + a_3 e_3 : a_1, a_2, a_3 \in [0,N) \big \}. \end{aligned}$$

Define

$$\begin{aligned} {{\mathbb {B}}_N}= \Big \{ \prod _{i=1}^3 [a_i,a_i+1) \subset {\mathbb {T}}_N: a_1,a_2,a_3 \in \{0,\dots ,N-1\} \Big \}. \end{aligned}$$

We call elements of ${{\mathbb {B}}_N}$ blocks. For any $B \subset {{\mathbb {B}}_N}$, we overload notation and write . Hence, $|B| = \int _B 1 dx$ is the number of blocks in B. In addition, we identify any $f \in {\mathbb {R}}^{{\mathbb {B}}_N}$ with the piecewise continuous function on ${\mathbb {T}}_N$ given by for .

Let $\phi \sim {\nu _{\beta ,N}}$. For any , let . Here, the integral is interpreted as the duality pairing between $\phi$ (a distribution) and the indicator function (a test function); we use this convention throughout. We let denote the block averaged field obtained from $\phi$.

Remark 3.1

Testing $\phi$ against , which is not smooth, yields a well-defined random variable on the support of ${\nu _{\beta ,N}}$. Indeed, $\phi$ belongs almost surely to $L^\infty$-based Besov spaces of regularity s for every $s < -\frac{1}{2}$ (see Appendix A for a review of Besov spaces and see Sect. 4 for the almost sure regularity of $\phi$). On the other hand, indicator functions of blocks belong to $L^1$-based Besov spaces of regularity s for every $s < 1$ or, more generally, $L^p$-based Besov spaces of regularity s for every $s < \frac{1}{p}$ (see, for example, Lemma 1.1 in [FR12]). This is sufficient to test $\phi$ against indicator functions of blocks (using e.g. Proposition A.1). We also give an alternative proof using a type of Itô isometry in Proposition 5.23.

3.2 Phase labels

We define a map ${\phi }\in {\mathbb {R}}^{{\mathbb {B}}_N}\mapsto \sigma \in \{-{\sqrt{\beta }}, 0, {\sqrt{\beta }}\}^{{\mathbb {B}}_N}$ called a phase label. A basic function of $\sigma$ is to identify whether the averages take values around the well at $+{\sqrt{\beta }}$, the well at $-{\sqrt{\beta }}$, or neither. We quantify this to a given precision $\delta \in (0,1)$, which is taken to be fixed in what follows.

• We say that is plus (resp. minus) valued if

  

  The set of plus (resp. minus) valued blocks is denoted ${\mathcal {P}}$ (resp. ${\mathcal {M}}$).

• The set of neutral blocks is defined as ${\mathcal {N}}= {{\mathbb {B}}_N}{\setminus } ({\mathcal {P}}\cup {\mathcal {M}})$.

Each block in ${{\mathbb {B}}_N}$ contains a midpoint. Given two distinct blocks in ${{\mathbb {B}}_N}$, we say that they are nearest-neighbours if their midpoints are of distance 1. They are $*$-neighbours if either they are nearest-neighbours or if their midpoints are of distance $\sqrt{3}$. For any , the $*$-connected ball centred at is the set consisting of and its $*$-neighbours. It contains exactly 27 blocks.

• We say that is plus good if every is plus valued. The set of plus good blocks is denoted ${\mathcal {P}}_G$.

• We say that is minus good if every is minus valued. The set of minus good blocks is denoted ${\mathcal {M}}_G$.

• The set of bad blocks is defined as ${\mathcal {B}}= {{\mathbb {B}}_N}{\setminus } ({\mathcal {P}}_G \cup {\mathcal {M}}_G)$.

Define the phase label $\sigma$ associated to ${\phi }$ of precision $\delta > 0$ by

The following proposition can be thought of as an extension of the contour bounds developed for $\phi ^4$ in 2D [GJS75, Theorem 1.2] to 3D.

Proposition 3.2

Let $\sigma$ be a phase label of precision $\delta \in (0,1)$. Then, there exists $\beta _0=\beta _0(\delta , \eta ) > 0$ and $C_P=C_P(\delta , \eta ) > 0$ such that, for $\beta > \beta _0$, the following holds for any $N \in 4{\mathbb {N}}$: for any set of blocks $B\subset {{\mathbb {B}}_N}$,

(3.1)

Proof

See Sect. 3.3.1. The main estimates required in the proof are given in Proposition 3.6, which extends [GJS75, Theorem 1.3] to 3D and improves the $\beta$-dependence. Assuming this, we then prove Proposition 3.2 in the spirit of the proof of [GJS75, Theorem 1.2]. $\quad \square$

3.3 Penalising bad blocks

Given a phase label, we partition the set of bad blocks ${\mathcal {B}}$ into two types.

• Frustrated blocks are blocks such that contains a neutral block. We denote the set of frustrated blocks ${\mathcal {B}}_F$.

• Interface block are blocks such that contains no neutral blocks, but there exists at least one pair of nearest-neighbours such that but . We denote the set of interface blocks ${\mathcal {B}}_I$.

For any and any nearest-neighbours , define:

(3.2)

Remark 3.3

Note that testing $:\phi ^2:$ against yields a well-defined random variable on the support of ${\nu _{\beta ,N}}$. We give a proof of this fact in Proposition 5.24.

We write for the set of unordered pairs of nearest-neighbour blocks in ${{\mathbb {B}}_N}$ such that . There are 54 elements in this set.

Lemma 3.4

Let $N \in {\mathbb {N}}$ and fix a phase label of precision $\delta \in (0,1)$. Then, for every ,

(3.3)

(3.4)

where $C_\delta = \min \Big ( \frac{\delta }{2}, 2-2\delta \Big ) > 0$.

Frustrated blocks are penalised by the potential ${\mathcal {V}}_\beta$ whereas interface blocks are penalised by the gradient term in the Gaussian measure. Lemma 3.4 formalises this through use of the random variables $Q_1, Q_2$ and $Q_3$, which (up to trivial modifications) were introduced in [GJS75]. $Q_1$ penalises frustrated blocks. $Q_2$ is an error term coming from the fact that the potential is written in terms of $\phi$ rather than ${\phi }$. $Q_3$ penalises interface blocks.

Proof of Lemma 3.4

For any ,

(3.5)

where in the penultimate line we have used that $\delta ^2 \leqslant \delta$.

By the definition of ${\mathcal {B}}_F$,

(3.6)

Using (3.5) applied to in (3.6) yields (3.3).

(3.4) is established by the following estimates: by the definition of ${\mathcal {B}}_I$,

$\square$

In order to use Lemma 3.4 to prove Proposition 3.2, we want to control expectations of $\cosh Q_1, \cosh Q_2$ and $\cosh Q_3$ by the exponentially small (in ${\sqrt{\beta }}$) prefactor in (3.3) and (3.4). Moreover, we want to control these expectations over a set of blocks as opposed to just single blocks.

Let $B_1, B_2 \subset {{\mathbb {B}}_N}$ and let $B_3$ be any set of unordered pairs of nearest-neighbours in ${{\mathbb {B}}_N}$. Define

(3.7)

Remark 3.5

Although the random variable does depend on the ordering of and , does not.

Proposition 3.6

For every $a_0 > 0$, there exist $\beta _0 = \beta _0(a_0,\eta )>0$ and $C_Q = C_Q(a_0,\beta _0,\eta )>0$ such that the following holds uniformly for all $\beta > \beta _0$, $a_1,a_2,a_3 \in {\mathbb {R}}$ such that $|a_i| \leqslant a_0$, and $N \in 4{\mathbb {N}}$: let $B_1, B_2 \subset {{\mathbb {B}}_N}$ and $B_3$ a set of unordered pairs of nearest-neighbour blocks in ${{\mathbb {B}}_N}$. Then,

$$\begin{aligned} \Big \langle \prod _{i=1}^3 \cosh \big (a_i Q_i(B_i)\big ) \Big \rangle _{\beta ,N} \leqslant e^{C_Q(|B_1|+|B_2|+|B_3|)} \end{aligned}$$

(3.8)

where $|B_3|$ is given by the number of pairs in $B_3$.

Proof

Proposition 3.6 is established in Sect. 6.3, but its proof takes up most of this article. The overall strategy is as follows: the crucial first step is to obtain upper and lower bounds on the free energy $-\log {\mathscr {Z}}_{\beta ,N}$ that are uniform in $\beta$ and extensive in the volume, $N^3$. We then build on this analysis to obtain upper bounds on expectations of the form $\langle \exp Q \rangle _{\beta ,N}$ that are uniform in $\beta$ and extensive in $N^3$. Here, Q is a placeholder for random variables that are derived from the $Q_i$’s, but that are supported on the whole of ${\mathbb {T}}_N$ rather than arbitrary unions of blocks. This is all done in Sect. 5, where the key results are Propositions 5.3 and 5.1 , within the framework developed in Sect. 4.

The next step in the proof is to use the chessboard estimates of Proposition 6.5 (which requires $N \in 4{\mathbb {N}}$) to bound the lefthand side of (3.8) in terms of $|B_1|+|B_2|+|B_3|$ products of expectations of the form $\langle \exp Q \rangle _{\beta ,N}^\frac{1}{N^3}$. Applying the results of Sect. 5 then completes the proof. $\quad \square$

Key features of the estimate (3.8) used in the proof of Proposition 3.2 are that it is uniform in $\beta$ and extensive in the support of the $Q_i$’s.

3.3.1 Proof of the Proposition 3.2 assuming Proposition 3.6

We first show that we can reduce to the case where B contains no $*$-neighbours, which simplifies the combinatorics later on. Identify ${{\mathbb {B}}_N}$ with a subset of ${\mathbb {Z}}^3$. For every $e_l \in \{-1,0,1\}^3$, let ${\mathbb {Z}}^3_l = e_l + (3 {\mathbb {Z}})^3$. There are 27 such sub-lattices which we order according to $l \in \{1,\dots ,27\}$. Note that ${\mathbb {Z}}^3 = \bigcup _{l=1}^{27} {\mathbb {Z}}^3_l$. Let ${\mathbb {B}}_N^l = {{\mathbb {B}}_N}\cap {\mathbb {Z}}^3_l$. Each $*$-connected ball in ${{\mathbb {B}}_N}$ contains at most one block from each of these ${\mathbb {B}}_N^l$.

Assume that (3.1) has been established for sets with no $*$-neighbours with constant $C_P'$. Then, by Hölder’s inequality,

(3.9)

thereby establishing (3.1) with $C_P = \frac{C_P'}{27}$.

Now assume that B contains no $*$-neighbours. Fix any $A \subset B$. Let and let . By our assumption, A contains no $*$-neighbours. Hence, for any there exists a unique such that ; we define the root of to be . Similarly, for any there exists a unique such that ; we define the root of to be . Note that the definition of root is A-dependent in both cases.

By Lemma 3.4, there exists $C_\delta$ such that

(3.10)

where the last sum is over all $A_1, A_2 \subset \mathrm {B}^*(A)$ and $A_3 \subset \mathrm {B}^*_{\mathrm {nn}}(B {\setminus } A)$ such that: no two blocks in $A_1 \cup A_2$ share a root, and no two pairs of blocks in $A_3$ share a root; and, $|A_1| + |A_2| = |A|$ and $|A_3| = |B {\setminus } A|$. We note that there are $(2 \cdot 27)^{|A|}=54^{|A|}$ possible $A_1$ and $A_2$, and $54^{|B {\setminus } A|}$ possible $A_3$.

By Proposition 3.6, there exists $C_Q$ such that, after taking expectations in (3.10) and using that $|A| + |B {\setminus } A| = |B|$, we obtain

Thus, choosing

$$\begin{aligned} {\sqrt{\beta }}> \frac{4\log 2 + 2\log 54 + 2C_Q}{C_\delta } \end{aligned}$$

yields (3.1) with $C_P= \frac{C_\delta }{2}$. This completes the proof.

3.4 Exchanging the block averaged field for the phase label

We now show that Propositions 3.2 and 3.6 allow one to reduce the problem of analysing the block averaged field to that of analysing the phase label. The main difficulty here is dealing with large fields, i.e. those ${\phi }$ for which $\int _{{\mathcal {B}}} |{\phi }|$ is large.

Proposition 3.7

Let $\delta , \delta ' \in (0,1)$ satisfy $\delta ' \leqslant \frac{\delta }{2}$. Then, there exists $\beta _0 = \beta _0(\delta ,\eta )>0$, $C=C(\delta , \beta _0, \eta )>0$ and $N_0=N_0(\delta )>0$ such that, for all $\beta > \beta _0$ and $N \in 4{\mathbb {N}}$ with $N>N_0$,

$$\begin{aligned} \frac{1}{N^3} \log {\nu _{\beta ,N}}\Big ( \int _{{\mathbb {T}}_N}\Big | \sigma - {\phi }\Big | dx > \delta {\sqrt{\beta }}N^3 \Big ) \leqslant -C{\sqrt{\beta }}\end{aligned}$$

(3.11)

where $\sigma$ is the phase label of precision $\delta ' \leqslant \frac{\delta }{2}$.

Proof

Observe that

$$\begin{aligned} \begin{aligned} {\nu _{\beta ,N}}\Bigg (&\int _{{\mathbb {T}}_N}| \sigma - {\phi }|dx> \delta {\sqrt{\beta }}N^3 \Bigg ) \\&\leqslant {\nu _{\beta ,N}}\Bigg ( \int _{{\mathbb {T}}_N}| \sigma - {\phi }|dx> \delta {\sqrt{\beta }}N^3 , |{\mathcal {B}}| < \frac{\delta }{8} N^3 \Bigg ) \\&\quad + {\nu _{\beta ,N}}\Bigg ( |{\mathcal {B}}| \geqslant \frac{\delta }{8} N^3 \Bigg ). \end{aligned} \end{aligned}$$

(3.12)

By Proposition 3.2, there exists $\beta _0>0$ and $C_P>0$ such that, for ${\sqrt{\beta }}> \max \Big (\sqrt{\beta _0}, \frac{16 \log 2}{C_P\delta } \Big )$,

$$\begin{aligned} \begin{aligned} {\nu _{\beta ,N}}\Bigg (|{\mathcal {B}}|\geqslant \frac{\delta }{8} N^3 \Bigg )&\leqslant \sum _{m = \lceil \frac{\delta }{8} N^3 \rceil }^{N^3} {\nu _{\beta ,N}}(|{\mathcal {B}}| = m) \\&\leqslant \sum _{m=\lceil \frac{\delta }{8} N^3 \rceil }^{N^3}{N^3 \atopwithdelims ()m}e^{-C_P{\sqrt{\beta }}m} \\&\leqslant 2^{N^3} e^{-\frac{C_P\delta }{8}{\sqrt{\beta }}N^3} \\&\leqslant e^{-\frac{C_P\delta }{16}{\sqrt{\beta }}N^3}. \end{aligned} \end{aligned}$$

(3.13)

Now consider the first term on the right hand side of (3.12). We decompose one step further:

$$\begin{aligned} {\nu _{\beta ,N}}\Bigg ( \int _{{\mathbb {T}}_N}| \sigma - {\phi }|dx> \delta {\sqrt{\beta }}N^3 , |{\mathcal {B}}| < \frac{\delta }{8} N^3 \Bigg ) \leqslant {\nu _{\beta ,N}}(T_1) + {\nu _{\beta ,N}}(T_2) \end{aligned}$$

where

$$\begin{aligned} T_1&= \Bigg \{\int _{{\mathbb {T}}_N}| \sigma - {\phi }|dx> \delta {\sqrt{\beta }}N^3 , \int _{{\mathcal {B}}} | {\phi }|dx \leqslant \frac{\delta }{2} {\sqrt{\beta }}N^3 \Bigg \} \\ T_2&= \Bigg \{ |{\mathcal {B}}| < \frac{\delta }{8} N^3, \int _{{\mathcal {B}}} | {\phi }|dx > \frac{\delta }{2} {\sqrt{\beta }}N^3 \Bigg \}. \end{aligned}$$

We show that $T_1 = \emptyset$ and that

$$\begin{aligned} {\nu _{\beta ,N}}(T_2) \leqslant e^{-C{\sqrt{\beta }}N^3} \end{aligned}$$

(3.14)

for some constant $C=C(\delta )>0$ and for $\beta$ sufficiently large. Combining these estimates with (3.13) completes the proof.

First, we treat $T_1$. On good blocks $|\phi _i - \sigma |$ is bounded by the ${\sqrt{\beta }}$ multiplied by the precision of the phase label ($\delta ' \leqslant \frac{\delta }{2}$ in this instance) and $\sigma = 0$ on bad blocks. Therefore, on the set $\Big \{ \int _{{\mathcal {B}}} |{\phi }| dx \leqslant \frac{\delta }{2} {\sqrt{\beta }}N^3\Big \}$, we have:

$$\begin{aligned} \int _{{\mathbb {T}}_N}|\sigma - {\phi }|dx&= \int _{{\mathcal {P}}_G \cup {\mathcal {M}}_G}|\sigma - {\phi }|dx + \int _{{\mathcal {B}}}|\sigma - {\phi }|dx \\&\leqslant \frac{\delta }{2} {\sqrt{\beta }}(|{\mathcal {P}}_G| + |{\mathcal {M}}_G|)+ \int _{{\mathcal {B}}}|{\phi }|dx \\&\leqslant \delta {\sqrt{\beta }}N^3 \end{aligned}$$

which shows that the first condition in $T_1$ is inconsistent with the second, so $T_1 = \emptyset$.

We turn our attention to $T_2$. Fix $B \subset {{\mathbb {B}}_N}$. By Chebyschev’s inequality, Young’s inequality, and Proposition 3.6, there exists $\beta _0>0$ and $C_Q>0$ such that, for $\beta > \beta _0$,

Therefore,

$$\begin{aligned} \begin{aligned} {\nu _{\beta ,N}}(T_2)&\leqslant \sum _{m = 1}^{\lfloor \frac{\delta }{8} N^3 \rfloor } \sum _{B:|B|=m} {\nu _{\beta ,N}}\Bigg ( \int _B |{\phi }| dx > \frac{\delta }{2} \beta N^3 \Bigg ) \\&\leqslant e^{-\frac{\delta }{2} {\sqrt{\beta }}N^3}\sum _{m=1}^{\lfloor \frac{\delta }{8} N^3 \rfloor } {N^3 \atopwithdelims ()m} e^{{\sqrt{\beta }}m}e^{(C_Q+\log 2)m} \\&\leqslant e^{-\frac{\delta }{2} {\sqrt{\beta }}N^3}2^{N^3} e^{\frac{\delta }{8}{\sqrt{\beta }}N^3}e^{\frac{(C_Q+\log 2)\delta }{8} N^3} \\&= e^{ \big ( - \frac{3\delta }{8} {\sqrt{\beta }}+ \log 2 + \frac{(C_Q+\log 2) \delta }{8} \big ) N^3}. \end{aligned} \end{aligned}$$

(3.15)

Taking

$$\begin{aligned} {\sqrt{\beta }}> \frac{16\log 2}{3\delta } + \frac{2}{3} (C_Q+\log 2) \end{aligned}$$

yields (3.14) with $C=\frac{3\delta }{16}$. $\quad \square$

3.5 Proof of the main result

Adapting an argument from [Bod02], we reduce the proof of Theorem 1.2 to bounding the probability that ${\phi }$ is far from $\pm {\sqrt{\beta }}$-valued functions on ${{\mathbb {B}}_N}$ whose boundary (between regions of opposite spins) is of certain fixed area. Proposition 3.7 then allows us to go from analysing ${\phi }$ to the phase label, for which we use existing results from [BIV00].

For any $B \subset {{\mathbb {B}}_N}$, let $\partial B$ denotes its boundary, which is given by the union of faces of blocks in B. Let $|\partial B| = \int _{\partial B} 1 ds(x)$, where ds(x) is the 2D Hausdorff measure (normalised so that faces have unit area). Thus, $|\partial B|$ is the number of faces in $\partial B$.

For any $a > 0$, let $C_a$ be the set of functions $f \in \{ \pm 1\}^{{\mathbb {B}}_N}$ such that $|\partial \{ f = +1\}|\leqslant aN^2$. For any $\delta > 0$, let ${\mathfrak {B}}(C_a,\delta )$ be the set of integrable functions g on ${\mathbb {T}}_N$ such that there exists $f \in C_a$ that satisfies $\int _{{\mathbb {T}}_N}|g-f| dx \leqslant \delta N^3$.

Proposition 3.8

Let $\delta , \delta ' \in (0,1)$ satisfy $\delta ' \leqslant \delta$. Then, there exists $\beta _0 = \beta _0(\delta ,\eta )>0$ and $C=C(\delta ,\beta _0,\eta )>0$ such that, for all $\beta > \beta _0$, the following estimate holds: for all $a>0$, there exists $N_0 = N_0(a,\delta ) \geqslant 4$ such that, for all $N > N_0$ dyadic,

$$\begin{aligned} \frac{1}{N^2} \log {\nu _{\beta ,N}}\Big ( \frac{1}{{\sqrt{\beta }}}\sigma \notin {\mathfrak {B}}(C_a, \delta ) \Big ) \leqslant -C{\sqrt{\beta }}a \end{aligned}$$

where $\sigma$ is the phase label of precision $\delta '$.

Proof

See [BIV00, Theorem 2.2.1] where Proposition 3.8 is proven for a more general class of phase labels that satisfy a Peierls’ type estimate such as the one in Proposition 3.2. We give a self-contained proof for our setting in Sect. 3.6. $\quad \square$

The following lemma is our main geometric tool. It is a weak form of the isoperimetric inequality on ${\mathbb {T}}_N$, although it can be reformulated in arbitrary dimension. Its proof is a standard application of Sobolev’s inequality and we include it for the reader’s convenience.

Lemma 3.9

There exists $C_I>0$ such that the following estimate holds for every $N \in {\mathbb {N}}$:

$$\begin{aligned} \min (|\{ f = 1 \}|, |\{ f = - 1 \}| ) \leqslant C_I |\partial \{ f = 1 \}|^\frac{3}{2} \end{aligned}$$

for every $f \in \{\pm 1 \}^{{\mathbb {B}}_N}$.

Proof

Let $\theta \in C^\infty _c({\mathbb {R}}^3)$ be rotationally symmetric with $\int _{{\mathbb {R}}^3} \theta dx = 1$. By Sobolev’s inequality, there exists C such that, for every $\varepsilon$,

$$\begin{aligned} \int _{{\mathbb {T}}_N}|f_\varepsilon - c_\varepsilon |^\frac{3}{2} dx \leqslant C \Big ( \int _{{\mathbb {T}}_N}|\nabla f_\varepsilon | dx \Big )^\frac{3}{2} \end{aligned}$$

(3.16)

where $f_\varepsilon = f *\varepsilon ^{-3} \theta ( \varepsilon ^{-1} \cdot )$ and $c_\varepsilon = \frac{1}{N^3} \int _{{\mathbb {T}}_N}f_\varepsilon dx$. Note that C is independent of N by scaling.

Letting $\varepsilon \rightarrow 0$ in the left hand side of (3.16), we obtain

$$\begin{aligned} \int _{{\mathbb {T}}_N}|f_\varepsilon - c_\varepsilon |^\frac{3}{2} dx \rightarrow \int _{{\mathbb {T}}_N}|f-c|^\frac{3}{2} dx \end{aligned}$$

(3.17)

where $c = \frac{ |\{ f = 1 \}| - |\{ f = -1 \}|}{N^3}$. Note that $c\in [-1,1]$.

Without loss of generality, assume $c\geqslant 0$. This implies that $|\{ f = 1 \}| \geqslant \{ f = -1 \}|$. Then, evaluating the integral on the righthand side of (3.17), we find that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {T}}_N}|f-c|^\frac{3}{2} dx&= (1-c)^\frac{3}{2} |\{f = 1\}| + (1+c)^\frac{3}{2} |\{ f = -1 \}| \\&= (1-c)^\frac{3}{2} cN^3 + \Big ( (1-c)^\frac{3}{2} + (1+c)^\frac{3}{2} \Big )|\{ f = -1\}| \\&\geqslant 2|\{f = -1 \}| \end{aligned} \end{aligned}$$

(3.18)

where we have used that the function

$$\begin{aligned} c \mapsto (1-c)^\frac{3}{2} + (1+c)^\frac{3}{2} \end{aligned}$$

has minimum at $c=0$ on the interval [0, 1].

For the term on the right hand side of (3.16), using duality we obtain

$$\begin{aligned} \int _{{\mathbb {T}}_N}|\nabla f_\varepsilon | dx = \sup _{{\mathbf {g}}\in C^\infty ({\mathbb {T}}_N,{\mathbb {R}}^3) : |{\mathbf {g}}|_\infty \leqslant 1} \Big | \int _{{\mathbb {T}}_N}\nabla f_\varepsilon \cdot {\mathbf {g}}dx \Big | \end{aligned}$$

(3.19)

where $|\cdot |_\infty$ denotes the supremum norm on $C^\infty ({\mathbb {T}}_N,{\mathbb {R}}^3)$.

For any such ${\mathbf {g}}$, using integration by parts and commuting the convolution with differentiation,

$$\begin{aligned} \Big | \int _{{\mathbb {T}}_N}\nabla f_\varepsilon {\mathbf {g}}dx \Big | = \Big | \int _{{\mathbb {T}}_N}f_\varepsilon \nabla \cdot {\mathbf {g}}dx \Big | = \Big | \int _{{\mathbb {T}}_N}f \nabla \cdot {\mathbf {g}}_\varepsilon dx \Big | \end{aligned}$$

(3.20)

where the ${\mathbf {g}}_\varepsilon$ is interpreted as convolving each component of ${\mathbf {g}}$ with $\varepsilon ^{-3} \theta (\varepsilon ^{-1}\cdot )$ separately.

Hence, by the divergence theorem, Young’s inequality for convolutions, and using the supremum norm bound on ${\mathbf {g}}$,

$$\begin{aligned} (3.20) = 2\Big | \int _{\partial \{ f = 1 \}} {\mathbf {g}}_\varepsilon \cdot {\hat{n}} ds(x) \Big | \leqslant 2 |\partial \{ f = 1\}| \end{aligned}$$

(3.21)

where ${\hat{n}}$ denotes the unit normal to $\partial \{ f = 1 \}$ pointing into $\{ f = -1\}$.

Inserting (3.21) in (3.19) implies that, for any $\varepsilon$,

$$\begin{aligned} \int _{{\mathbb {T}}_N}|\nabla f_\varepsilon | dx \leqslant 2 |\partial \{ f = 1 \}|. \end{aligned}$$

(3.22)

Thus, by inserting (3.22), (3.17) and (3.18) into (3.16), we obtain

$$\begin{aligned} |\{ f = -1 \}| \leqslant \sqrt{2} C |\partial \{ f = 1 \}|^\frac{3}{2}. \end{aligned}$$

$\square$

Proof (Proof of Theorem 1.2)

Let $\zeta \in (0,1)$. Choose $a>0$ and $\delta \in (0,1)$ such that

$$\begin{aligned} 1 - 2 C_I a^\frac{3}{2} - \delta = \zeta \end{aligned}$$

(3.23)

where $C_I$ is the same constant as in Lemma 3.9. We first show that

$$\begin{aligned} \{ {\mathfrak {m}}_N(\phi ) \in (-\zeta {\sqrt{\beta }}, \zeta {\sqrt{\beta }}) \} \subset \Big \{ \frac{1}{{\sqrt{\beta }}} {\phi }\notin {\mathfrak {B}}(C_a,\delta ) \Big \}. \end{aligned}$$

(3.24)

Assume $\frac{1}{{\sqrt{\beta }}} {\phi }\in {\mathfrak {B}}(C_a,\delta ).$ Then, there exists $f \in C_a$ such that

$$\begin{aligned} \int _{{\mathbb {T}}_N}\Big | \frac{1}{{\sqrt{\beta }}}{\phi }- f \Big |dx \leqslant \delta N^3. \end{aligned}$$

This implies

$$\begin{aligned} \Biggr | \Big | \int _{{\mathbb {T}}_N}\frac{1}{{\sqrt{\beta }}} {\phi }dx\Big | - \Big | \int _{{\mathbb {T}}_N}f dx \Big |\Biggr | \leqslant \delta N^3 \end{aligned}$$

from which we deduce, together with Lemma 3.9,

$$\begin{aligned} \Big |\frac{1}{{\sqrt{\beta }}}{\mathfrak {m}}_N(\phi ) \Big |&\geqslant 1 - \frac{2\min \Big ( |\{f=+1\}|, |\{f=-1\}|\Big )}{N^3} -\delta . \\&\geqslant 1 - \frac{2C_I |\partial \{ f =+1\}|^\frac{3}{2}}{N^3}- \delta . \end{aligned}$$

Since $f \in C_a$, we obtain

$$\begin{aligned} |{\mathfrak {m}}_N(\phi )| \geqslant {\sqrt{\beta }}(1-2C_Ia^\frac{3}{2} - \delta ) = \zeta {\sqrt{\beta }}\end{aligned}$$

by (3.23).

Hence,

$$\begin{aligned} \Big \{ \frac{1}{{\sqrt{\beta }}}{\phi }\in {\mathfrak {B}}(C_a,\delta ) \Big \} \subset \{ |{\mathfrak {m}}_N(\phi )| \geqslant \zeta {\sqrt{\beta }}\}. \end{aligned}$$

Taking complements establishes (3.24).

Now let $\sigma$ be the phase label of precision $\frac{\delta }{2}$. Note that

$$\begin{aligned} \Big \{ \frac{1}{{\sqrt{\beta }}}{\phi }\notin {\mathfrak {B}}(C_a,\delta ) \Big \} \subset \Big \{ \frac{1}{{\sqrt{\beta }}}\sigma \notin {\mathfrak {B}}(C_a, 2\delta ) \Big \} \bigcup \Big \{ \int _{{\mathbb {T}}_N}| {\phi }- \sigma |dx>\delta {\sqrt{\beta }}N^3\Big \}. \end{aligned}$$

Applying Proposition 3.7, Proposition 3.8, and using (3.24) finishes the proof. $\quad \square$

3.6 Proof of Proposition 3.8

For any $B \subset {{\mathbb {B}}_N}$, let $\partial ^* B$ be the set of blocks in B with $*$-neighbours in ${\mathbb {T}}_N{\setminus } B$. Note that this is not the same as $\partial B$, which was defined earlier. Let ${\mathcal {D}}$ be the set of $*$-connected components of $\partial ^* ({\mathbb {T}}_N{\setminus } {\mathcal {M}}_G)$. We call this the set of defects. Necessarily, any $\Gamma \in {\mathcal {D}}$ satisfies $\Gamma \subset {\mathcal {B}}$.

Fix $\gamma \in (0, 1)$. Let ${\mathcal {D}}^\gamma \subset {\mathcal {D}}$ be the set of $\Gamma \in {\mathcal {D}}$ such that $|\Gamma | \leqslant 6 N^\gamma$. The elements of ${\mathcal {D}}^\gamma$ are called $\gamma$-small defects and the elements of ${\mathcal {D}}{\setminus } {\mathcal {D}}^\gamma$ are called $\gamma$-large defects.

Take any $\Gamma \in {\mathcal {D}}^\gamma$. Recall that we identify $\Gamma$ with the subset of ${\mathbb {T}}_N$ given by the union of blocks in $\Gamma$. Write $\mathrm {Cl}(\Gamma )$ for its closure in ${\mathbb {T}}_N$. The condition $\gamma < 1$ ensures that, provided N is taken sufficiently large depending on $\gamma$, any $\Gamma \in {\mathcal {D}}^\gamma$ is contained in a (translate of a) sphere of radius $\frac{N}{4}$ in ${\mathbb {T}}_N$. Let $\mathrm {Ext}(\Gamma )$ be the unique connected component of ${\mathbb {T}}_N{\setminus } \mathrm {Cl}(\Gamma )$ that intersects with the complement of this sphere. Let $\mathrm {Int}(\Gamma ) = {\mathbb {T}}_N{\setminus } \mathrm {Ext}(\Gamma )$. We identify $\mathrm {Ext}(\Gamma )$ and $\mathrm {Int}(\Gamma )$ with their representations as subsets of ${{\mathbb {B}}_N}$. Note that $\Gamma \subset \mathrm {Int}(\Gamma )$ and generically the inclusion strict, e.g. when $\Gamma$ encloses a region.

Let ${\mathcal {D}}^{\gamma ,\max }$ be the set of $\Gamma \in {\mathcal {D}}^{\gamma }$ such that $\Gamma \bigcap \mathrm {Int}({\tilde{\Gamma }}) = \emptyset$ for any ${\tilde{\Gamma }} \in {\mathcal {D}}^\gamma {\setminus } \{ \Gamma \}$. In other words, ${\mathcal {D}}^{\gamma ,\max }$ is the set of $\gamma$-small defects that are not contained in the interior of any other $\gamma$-small defects, and we call these maximal $\gamma$-small defects.

We define two events, one corresponds to the total surface area of $\gamma$-large defects being small and the other corresponding to the total volume contained within maximal $\gamma$-small defects being small. Let

$$\begin{aligned} S_1&= \Bigg \{ \sum _{\Gamma \in {\mathcal {D}}{\setminus } {\mathcal {D}}^\gamma } |\Gamma | \leqslant \frac{a}{6}N^2 \Bigg \} \\ S_2&= \Bigg \{ \sum _{\Gamma \in {\mathcal {D}}^{\gamma ,\max }} |\mathrm {Int}(\Gamma )| \leqslant \frac{\delta }{4} N^3 \Bigg \}. \end{aligned}$$

We now show that for $\phi \in S_1 \cap S_2 \cap \{|{\mathcal {B}}| < \frac{\delta }{2} N^3\}$, we have $\frac{1}{{\sqrt{\beta }}}\sigma \in {\mathfrak {B}}(C_a,\delta )$.

We obtain a $\pm {\sqrt{\beta }}$-valued spin configuration from $\sigma$ by erasing all $\gamma$-small defects in two steps: First, we reset the values on bad blocks to ${\sqrt{\beta }}$. Define $\sigma _1 \in \{ \pm {\sqrt{\beta }}\}^{{\mathbb {B}}_N}$ by if , otherwise . Second, define $\sigma _2\in \{\pm {\sqrt{\beta }}\}^{{\mathbb {B}}_N}$ as follows: Given for some $\Gamma \in {\mathcal {D}}^{\gamma ,\max }$, let , where is any block in $\mathrm {Ext}(\Gamma )$ that is $*$-neighbours with a block in $\Gamma$. Note that the second step is well-defined since the first step ensures that every block in $\mathrm {Ext}(\Gamma )$ that is $*$-neighbours with $\Gamma$ has the same value. See Figure 1 for an example of this procedure.

Fig. 1

An example of the $\sigma$ to $\sigma _2$ procedure (left to right). Image courtesy of J. N. Gunaratnam

From the definition of $S_1$ and using that the factor 6 in the definition of $\gamma$-small defects accounts for the discrepancy between $|\partial \cdot |$ and $|\partial ^*\cdot |$,

$$\begin{aligned} |\partial \{ \sigma _2 = +{\sqrt{\beta }}\}| \leqslant aN^2 \end{aligned}$$

yielding $\frac{1}{{\sqrt{\beta }}}\sigma _2 \in C_a$. Then, from the definition of $S_2$ and using the smallness assumption on the number of bad blocks,

$$\begin{aligned} \int _{{\mathbb {T}}_N}\frac{1}{{\sqrt{\beta }}}\Big |\sigma - \sigma _2 \Big |dx \leqslant 2\sum _{\Gamma \in {\mathcal {D}}^{\gamma ,\max } } |\mathrm {Int}(\Gamma )| + |{\mathcal {B}}|< 2 \frac{\delta }{4} N^3 + \frac{\delta }{2} N^3 < \delta N^3 \end{aligned}$$

which establishes that $\frac{1}{{\sqrt{\beta }}}\sigma \in {\mathfrak {B}}(C_a,\delta )$.

We deduce that the event $\Big \{ \frac{1}{{\sqrt{\beta }}}\sigma \not \in {\mathfrak {B}}(C_a,\delta ) \Big \}$ necessarily implies one of three things: either there are many bad blocks; or, the total surface area of $\gamma$-large defects is large; or, the density of $\gamma$-small defects is high. That is,

$$\begin{aligned} \begin{aligned} {\nu _{\beta ,N}}\Big ( \frac{1}{{\sqrt{\beta }}}\sigma \notin&{\mathfrak {B}}(C_a,\delta ) \Big ) \\&\leqslant {\nu _{\beta ,N}}\Big (|{\mathcal {B}}|\geqslant \frac{\delta }{2} N^3\Big ) + {\nu _{\beta ,N}}(S_1^c) + {\nu _{\beta ,N}}(S_2^c). \end{aligned} \end{aligned}$$

(3.25)

Proposition 3.2 gives control on the first event. The other two are controlled by the following lemmas.

Lemma 3.10

Let $\gamma ,\delta \in (0,1)$. Then, there exists $\beta _0=\beta _0(\gamma ,\delta ,\eta )>0$ and $C=C(\gamma ,\delta ,\beta _0,\eta )>0$ such that, for all $\beta > \beta _0$, the following holds: for any $a > 0$, there exists $N_0 = N_0(\gamma ,a)>0$ such that, for any $N \in 4{\mathbb {N}}$ with $N > N_0$,

$$\begin{aligned} \frac{1}{N^2}\log {\nu _{\beta ,N}}\Bigg ( \sum _{\Gamma \in {\mathcal {D}}{\setminus } {\mathcal {D}}^\gamma }|\Gamma | > a N^2 \Bigg ) \leqslant -C{\sqrt{\beta }}\Big ( a + \frac{N^\gamma }{N^2} \Big ) \end{aligned}$$

where the underlying phase label is of precision $\delta$.

Proof

We give a proof based on arguments from [DKS92, Theorem 6.1] in Sect. 3.6.1. $\quad \square$

Lemma 3.11

Let $\gamma ,\delta ,\delta ' \in (0,1)$. Then, there exists $\beta _0=\beta _0(\gamma ,\delta ,\delta ',\eta )>0$, $C=C(\gamma ,\delta , \delta ',\beta _0,\eta )>0$ and $N_0=N_0(\gamma , \delta ) \geqslant 4$ such that, for all $\beta > \beta _0$ and $N > N_0$ dyadic,

$$\begin{aligned} \frac{1}{N^2} \log \nu _{\beta ,N} \Bigg ( \sum _{\Gamma \in {\mathcal {D}}^{\gamma ,\max }} |\mathrm {Int}(\Gamma )| > \delta N^3 \Bigg ) \leqslant -C{\sqrt{\beta }}\frac{N}{N^{3\gamma }} \end{aligned}$$

where the underlying phase label is of precision $\delta '$.

Proof

See [BIV00, Section 5.1.3] for a proof in a more general setting. We give an alternative proof in Sect. 3.6.2 that avoids the use of techniques from percolation theory. $\quad \square$

As in (3.13), by Proposition 3.2 there exists $C_P > 0$ such that

$$\begin{aligned} {\nu _{\beta ,N}}(|{\mathcal {B}}|\geqslant \delta N^3) \leqslant e^{-\frac{C_P\delta }{4}{\sqrt{\beta }}N^3} \end{aligned}$$

(3.26)

provided ${\sqrt{\beta }}> \frac{4\log 2}{\delta C_P}$.

Therefore, from (3.25), (3.26), Lemma 3.10 and Lemma 3.11, there exists $C>0$ such that

$$\begin{aligned} \frac{1}{N^2}\log {\nu _{\beta ,N}}\Big ( \sigma \notin {\mathfrak {B}}(C_a,\delta ) \Big ) \leqslant -C{\sqrt{\beta }}\min \Big ( N, a + \frac{N^\gamma }{N^2}, \frac{N}{N^{3\gamma }}\Big ). \end{aligned}$$

Taking $\gamma < \frac{1}{3}$ and N sufficiently large completes the proof. All that remains is to show Lemmas 3.10 and 3.11.

3.6.1 Proof of Lemma 3.10

By a union bound

$$\begin{aligned} \begin{aligned} {\nu _{\beta ,N}}\Biggr ( \sum _{\Gamma \in {\mathcal {D}}{\setminus } {\mathcal {D}}^\gamma }|\Gamma |> a N^2 \Biggr )&= \sum _{\begin{array}{c} \{ \Gamma _i \}: |\Gamma _i|> N^\gamma \\ \sum _i |\Gamma _i|> aN^2 \end{array} }{\nu _{\beta ,N}}\Big ({\mathcal {D}}{\setminus }{\mathcal {D}}^\gamma = \{ \Gamma _i \} \Big ) \\&\leqslant \sum _{\begin{array}{c} \{ \Gamma _i \}: |\Gamma _i|> N^\gamma \\ \sum _i |\Gamma _i| > aN^2 \end{array} } {\nu _{\beta ,N}}\Big ( \Gamma _i \subset {\mathcal {B}}\text { for all } \Gamma _i \in \{ \Gamma _i \} \Big ), \end{aligned} \end{aligned}$$

(3.27)

where$\{ \Gamma _i \}$ refers to a non-empty set of distinct $*$-connected subsets of ${{\mathbb {B}}_N}$.

By Proposition 3.2 there exists $C_P$ such that, for any $\{ \Gamma _i \}$,

Inserting this into (3.27) and using the trivial estimate $\sum |\Gamma _i| \geqslant \frac{1}{2} aN^2 + \frac{1}{2} \sum |\Gamma _i|$,

$$\begin{aligned} \begin{aligned} {\nu _{\beta ,N}}\Biggr ( \sum _{\Gamma \in {\mathcal {D}}{\setminus }{\mathcal {D}}^\gamma }|\Gamma |> a N^2 \Biggr )&\leqslant \sum _{\begin{array}{c} \{ \Gamma _i \}: |\Gamma _i|> N^\gamma \\ \sum _i |\Gamma _i|> aN^2 \end{array} } e^{-C_P{\sqrt{\beta }}\sum |\Gamma _i|} \\&\leqslant e^{-\frac{C_P}{2} {\sqrt{\beta }}aN^2} \sum _{\{\Gamma _i\}: |\Gamma _i|>N^\gamma } e^{-\frac{C_P}{2} {\sqrt{\beta }}\sum |\Gamma _i|} \\&= e^{-\frac{C_P}{2} {\sqrt{\beta }}aN^2} \sum _{\{ \Gamma _i \}: |\Gamma _i| > N^\gamma } \prod _{\Gamma _i \in \{ \Gamma _i \}} e^{-\frac{C_P}{2} {\sqrt{\beta }}|\Gamma _i|}. \end{aligned} \end{aligned}$$

(3.28)

Summing first over the number of elements in $\{ \Gamma _i \}$ and then the number of $*$-connected regions containing a fixed number of blocks,

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \{ \Gamma _i \} \\ |\Gamma _i|>N^\gamma \end{array}} \prod _{\Gamma _i \in \{ \Gamma _i \}} e^{-\frac{C_P}{2} {\sqrt{\beta }}|\Gamma _i|}&= \sum _{m=1}^\infty \sum _{\{\Gamma _i\}_{i=1}^m : |\Gamma _i| > N^\gamma } \prod _{i=1}^m e^{-\frac{C_P}{2} {\sqrt{\beta }}|\Gamma _i|} \\&\leqslant \sum _{m=1}^\infty \Big ( \sum _{\Gamma \text { *-connected }: |\Gamma | \geqslant N^\gamma } e^{-\frac{C_P}{2} {\sqrt{\beta }}|\Gamma |} \Big )^m \\&\leqslant \sum _{m=1}^\infty \Big ( \sum _{n \geqslant N^\gamma } N^3 27 \cdot 26^{n-1} e^{-\frac{C_P}{2}{\sqrt{\beta }}n} \Big )^m \\&\leqslant \sum _{m=1}^\infty e^{3m \log N - \frac{C_P}{4} {\sqrt{\beta }}m N^\gamma } \Big ( \sum _{n \geqslant 1} e^{-\frac{C_P}{4} {\sqrt{\beta }}n} \Big )^m \\&\leqslant \sum _{m=1}^\infty e^{\Big (3\log N - \frac{C_P}{4} {\sqrt{\beta }}N^\gamma \Big )m} \\&\leqslant e^{-\frac{C_P}{8}{\sqrt{\beta }}N^\gamma }\sum _{m=1}^\infty e^{3m\log N - \frac{C_P}{8} {\sqrt{\beta }}m N^\gamma } \end{aligned} \end{aligned}$$

(3.29)

provided ${\sqrt{\beta }}> \max \Big ( \frac{4\log 27}{C_P}, \frac{4\log 2}{C_P} \Big ) = \frac{4 \log 27}{C_P}$ (note that the condition arises so that $e^{-\frac{C_P}{4} {\sqrt{\beta }}} < \frac{1}{2}$, so that the geometric series with this rate is bounded by 1).

For any $\gamma > 0$, the final series in (3.29) is summable provided $N^\gamma > \log N$ and ${\sqrt{\beta }}> \frac{24}{C_P}$, thereby finishing the proof.

3.6.2 Proof of Lemma 3.11

Choose $2 N^\gamma \leqslant K \leqslant 4 N^\gamma$ such that K divides N. Such a choice is possible since we take N to be a sufficiently large dyadic. Let

Elements of ${\mathbb {B}}_N^K$ are called K-blocks.

We say that two distinct K-blocks are $*_K$-neighbours if their corresponding midpoints are of distance at most $K\sqrt{3}$. We define the $*_K$-connected ball around to be the set containing itself and its $*_K$-neighbours. As in the proof of Proposition 3.2, we can decompose ${\mathbb {B}}_N^K = \bigcup _{l=1}^{27} {\mathbb {B}}_N^{K,l}$ such that any $*_K$-connected ball in ${\mathbb {B}}_N^K$ contains exactly one K-block from each element of the decomposition.

For each , distinguish the unit block . For every $h \in \{0,\dots ,K-1\}^3$, let $\tau _h$ be the translation map on ${{\mathbb {B}}_N}$ induced from the translation map on ${\mathbb {T}}_N$. We identify . Denote the set of distinguished unit blocks in ${\mathbb {B}}_N^K$ (respectively, ${\mathbb {B}}_N^{K,l}$) as ${\mathbb {U}}{\mathbb {B}}_N^K$ (respectively, ${\mathbb {U}}{\mathbb {B}}_N^{K,l}$).

By our choice of K, $\mathrm {Int}(\Gamma )$ is entirely contained in a translation of a K-block for any $\Gamma \in {\mathcal {D}}^\gamma$. As a result, $\mathrm {Int}(\Gamma )$ intersects at most one K-block in ${\mathbb {B}}_N^{K,l}$ for any fixed l.

Using the correspondence between K-blocks and unit blocks described above, we have

Hence,

(3.30)

where the maximum is over $h \in \{0,\dots ,K-1\}^3$ and $1\leqslant l \leqslant 27$.

Let $E_k$ be the event that precisely k indicator functions appearing on the right hand side of (3.30) are nonzero. In other words, $E_k$ is the event that there are k distinct defects of size at most $N^\gamma$ such that the k distinct , where , are contained in their interiors.

Given a block there are $27 \cdot 26^{n-1}$ possible defects of size n that contain this block. Thus, by Proposition 3.2, there exists $C_P$ such that

$$\begin{aligned} {\nu _{\beta ,N}}(E_k)&\leqslant { \frac{N^3}{27K^3} \atopwithdelims ()k } \sum _{1 \leqslant n_1, \dots , n_k \leqslant N^\gamma } \prod _{j=1}^k n_j\cdot 26\cdot 27^{n_j-1}e^{-C_P{\sqrt{\beta }}n_j} \nonumber \\&\leqslant { \frac{N^3}{27K^3} \atopwithdelims ()k } e^{-\frac{C_P}{2} {\sqrt{\beta }}k} \Big (\sum _{n=1}^{N^\gamma } n \cdot 26 \cdot 27^{n-1} e^{-\frac{C_P}{2} {\sqrt{\beta }}n} \Big )^k \nonumber \\&\leqslant { \frac{N^3}{27K^3} \atopwithdelims ()k } e^{- \frac{C_P}{2} {\sqrt{\beta }}k} \end{aligned}$$

(3.31)

provided e.g. ${\sqrt{\beta }}> \max \Big ( \frac{4\log 27}{C_P}, \frac{2\log 2}{C_P} \Big ) = \frac{4\log 27}{C_P}$. This estimate is uniform over the choice of h and l.

By a union bound on (3.30), using (3.31), and that $2N^\gamma \leqslant K \leqslant 4N^\gamma$,

$$\begin{aligned} {\nu _{\beta ,N}}\Big ( \sum _{\Gamma \in {\mathcal {D}}^{\gamma ,\max }} |\mathrm {Int}(\Gamma )|>\delta N^3)&\leqslant 27 K^3 \sum _{k = \lfloor \frac{\delta N^3}{27 K^3} \rfloor + 1}^{\frac{N^3}{27K^3}}{ \frac{N^3}{27K^3} \atopwithdelims ()k } e^{-\frac{C_P}{2} {\sqrt{\beta }}k} \\&\leqslant 27K^3 \cdot 2^{\frac{N^3}{27K^3}}e^{-\frac{\delta C_P}{2\cdot 27}{\sqrt{\beta }}\frac{N^3}{K^3}} \\&\leqslant 27 \cdot 64 e^{3 \gamma \log N + \frac{\log 2}{27 \cdot 8}\frac{N^3}{N^{3\gamma }} - \frac{\delta C_P}{27 \cdot 16}{\sqrt{\beta }}\frac{N^3}{N^{3\gamma }}} \\&\leqslant 27 \cdot 64 e^{-\frac{\delta C_P}{27 \cdot 32}{\sqrt{\beta }}\frac{N^3}{N^{3\gamma }}} \end{aligned}$$

provided $\gamma \log N < N^{3-3\gamma }$ and ${\sqrt{\beta }}> \frac{81 \cdot 32 + 4\log 2}{\delta C_P}$. Taking logarithms and dividing by $N^2$ completes the proof.

4 Boué–Dupuis Formalism for $\phi ^4_3$

In this section we introduce the underlying framework that we build on to analyse expectations of certain random variables under ${\nu _{\beta ,N}}$, as required in the proof of Proposition 3.6. This framework was originally developed in [BG19] to show ultraviolet stability for $\phi ^4_3$ and identify its Laplace transform.

In particular, we want to obtain estimates on expectations of the form $\langle e^{Q_K} \rangle _{\beta ,N,K}$, where $Q_K$ are random variables that converge (in an appropriate sense) to some random variable Q of interest. We always work with a fixed ultraviolet cutoff K and establish estimates on $\langle e^{Q_K} \rangle _{\beta ,N,K}$ that are uniform in K: this requires handling of ultraviolet divergences. The first observation is that we can represent such expectations as a ratio of Gaussian expectations:

$$\begin{aligned} \langle e^{Q_K} \rangle _{\beta ,N,K} = \frac{{\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K) + Q_K(\phi _K)}}{{\mathscr {Z}}_{\beta ,N,K}} \end{aligned}$$

(4.1)

where we recall ${\mathbb {E}}_N$ denotes expectation with respect to $\mu _N$ and ${\mathscr {Z}}_{\beta ,N,K} = {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K)}$ is the partition function.

We then introduce an auxiliary time variable that continuously varies the ultraviolet cutoff between 0 and K, and use it to represent these Gaussian expectations in terms of expectations of functionals of finite dimensional Brownian motions. This allows us to use the Boué–Dupuis variational formula given in Proposition 4.7 to write these expectations in terms of a stochastic control problem. Hence, the problem of obtaining bounds is translated into choosing appropriate controls. An insight made in [BG19] is that one can use methods developed in the context of singular stochastic PDEs, specifically the paracontrolled calculus approach of [GIP15], within the control problem to kill ultraviolet divergences.

Remark 4.1

In the following, we make use of tools in Appendices A and A concerning Besov spaces and paracontrolled calculus. In addition, for the rest of Sects. 4 and 5 , we consider $N \in {\mathbb {N}}$ fixed and drop it from notation when clear.

4.1 Construction of the stochastic objects

Fix $\kappa _0 > 0$ sufficiently small. We equip $\Omega = C({\mathbb {R}}_+; {\mathcal {C}}^{-\frac{3}{2} -\kappa _0})$ with its Borel $\sigma$-algebra. Denote by ${\mathbb {P}}$ the probability measure on $\Omega$ under which the coordinate process $X_{\bullet }=(X_k)_{k \geqslant 0}$ is an $L^2$ cylindrical Brownian motion. We write ${\mathbb {E}}$ to denote expectation with respect to ${\mathbb {P}}$. We consider the filtered probability space $(\Omega , {\mathcal {A}}, ({\mathcal {A}}_k)_{k \geqslant 0},{\mathbb {P}})$, where ${\mathcal {A}}$ is the ${\mathbb {P}}$-completion of the Borel $\sigma$-algebra on $\Omega$, and $({\mathcal {A}}_k)_{k \geqslant 0}$ is the natural filtration induced by X and augmented with ${\mathbb {P}}$-null sets of ${\mathcal {A}}$.

Given $n \in (N^{-1}{\mathbb {Z}})^3$, define the process $B^n_\bullet$ by $B^n_k = \frac{1}{N^\frac{3}{2}} \int _{{\mathbb {T}}_N}X_k e_{-n} dx$, where $e_n(x) = e^{2\pi i n \cdot x}$ and we recall that the integral denotes duality pairing between distributions and test functions. Then, $\{B^n_\bullet : n \in (N^{-1}{\mathbb {Z}})^3\}$ is a set of complex Brownian motions defined on $(\Omega , {\mathcal {A}}, ({\mathcal {A}}_k)_{k \geqslant 0},{\mathbb {P}})$, independent except for the constraint $\overline{B_k^n} = B_k^{-n}$. Moreover,

$$\begin{aligned} X_k = \frac{1}{N^3} \sum _{n \in (N^{-1}{\mathbb {Z}})^3} B_k^n N^\frac{3}{2} e_n \end{aligned}$$

where ${\mathbb {P}}$-almost surely the sum converges in ${\mathcal {C}}^{-\frac{3}{2} - \kappa _0}$.

Let ${\mathcal {J}}_k$ be the Fourier multiplier with symbol

$$\begin{aligned} {\mathcal {J}}_k(\cdot ) = \frac{\sqrt{\partial _k \rho _k^2( \cdot )}}{\langle \cdot \rangle } \end{aligned}$$

where $\rho _k$ is the ultraviolet cutoff defined in Sect. 2 and we recall $\langle \cdot \rangle = \sqrt{\eta + 4\pi ^2|\cdot |^2}$. ${\mathcal {J}}_k$ arises from a continuous decomposition of the covariance of the pushforward measure $\mu _N$ under $\rho _k$:

$$\begin{aligned} \int _0^k {\mathcal {J}}_{k'}^2(\cdot ) d{k'} = \frac{\rho _k^2(\cdot )}{\langle \cdot \rangle ^2} = {\mathcal {F}}\Big \{ {\mathcal {F}}^{-1}(\rho _k) *(-\Delta + \eta )^{-1} *{\mathcal {F}}^{-1}(\rho _k) \Big \} (\cdot ) \end{aligned}$$

where ${\mathcal {F}}$ denotes the Fourier transform and ${\mathcal {F}}^{-1}$ denotes its inverse (see “Appendix A”). Note that the function $\partial _k \rho _k^2$ has decay of order $\langle k \rangle ^{-\frac{1}{2}}$ and the corresponding multiplier is supported frequencies satisfying $|n| \in (c_\rho k, C_\rho k)$ for some $c_\rho < C_\rho$. Thus, we may think of ${\mathcal {J}}_k$ as having the same regularising properties as the multiplier ; precise statements are given in Proposition A.9.

Define the process by

(4.2)

is a centred Gaussian process with covariance:

for any $f,g \in L^2$. Thus, the law of is the law of $\rho _k \phi$ where $\phi \sim \mu _N$. As with other processes in the following, we simply write .

4.1.1 Renormalised multilinear functions of the free field

The second, third, and fourth Wick powers of are the space-stationary stochastic processes defined by:

where we recall from Sect. 2 that . Note that , and are equal in law to $:\phi _k^2:, :\phi _k^3:$, and $:\phi _k^4:$, respectively.

The Wick powers of can be expressed as iterated integrals using Itô’s formula (see [Nua06, Section 1.1.2]). We only need the iterated integral representation :

(4.3)

where we have used the convention that sums over frequencies $n_i$ range over $(N^{-1}{\mathbb {Z}})^3$.

We define additional space-stationary stochastic processes by

We make two observations: first, a straightforward calculation shows that diverges in variance as $k \rightarrow \infty$. However, due to the presence of ${\mathcal {J}}_k$, can be made sense of as $k \rightarrow \infty$. See Lemma 4.6.

Second, , , and are renormalised resonant products of , , and , respectively. The latter products are classically divergent in the limit $k \rightarrow \infty$. We refer to Remark 4.2 for an explanation of why the resonant product is used.

Remark 4.2

Let $f \in {\mathcal {C}}^{s_1}$ and $g \in {\mathcal {C}}^{s_2}$ for $s_1< 0 < s_2$. Bony’s decomposition states that, if the product exists, and is of regularity $s_1$ (see Appendix A). Since paraproducts are always well-defined (see Proposition A.5), the resonant product contains all of the difficulty in defining the product. However, the resonant product gives regularity information of order $s_1 + s_2$ (see Proposition A.6), which is strictly stronger than the regularity information of the product: i.e. the bound on is strictly stronger than the bound on $\Vert fg \Vert _{{\mathcal {C}}^{s_1}}$. This is the key property that makes paracontrolled calculus useful in this context [GIP15].

The required renormalisations of and are related to the usual “sunset” diagram appearing in the perturbation theory for $\phi ^4_3$,

(4.4)

See [Fel74, Theorem 1]. We emphasise that depends on $\eta , N$ and k.

By the fundamental theorem of calculus, the Leibniz rule, and symmetry,

Thus, the renormalisations of and are given by and , respectively.

Remark 4.3

It is straightforward to verify that there exists $C=C(\eta )>0$ such that

Let . We refer to the coordinates of $\Xi$ as diagrams. The following proposition gives control over arbitrarily high moments of diagrams in Besov spaces.

Proposition 4.4

For any $p,p' \in [1,\infty )$, $q \in [1,\infty ]$, and $\kappa > 0$ sufficiently small, there exists $C=C(p,p',q,\kappa ,\eta )>0$ such that

(4.5)

Proof

See [BG19, Lemma 24]. $\quad \square$

Remark 4.5

The constant on the righthand side of (4.5) is independent of N because our Besov spaces are defined with respect to normalised Lebesgue measure (see Appendix A). For $p=\infty$, bounds that are uniform in N do not hold. Indeed, for $L^\infty$-based norms, there is in general no chance of controlling space-stationary processes uniformly in the volume. Thus, we cannot work in Besov-Hölder spaces.

We prove the bound in (4.5) for since it illustrates the role of ${\mathcal {J}}_k$, is used later in the proof of Proposition 5.23, and gives the reader a flavour of how to prove the bounds on the other diagrams.

Lemma 4.6

There exists $C=C(\eta )>0$ such that, for any $n \in (N^{-1}{\mathbb {Z}})^3$,

(4.6)

As a consequence, for every $p \in [1,\infty )$ and $s < \frac{1}{2}$, there exists $C=C(p,s,\eta )>0$ such that

Proof

Inserting (4.3) in the definition of and switching the order of integration,

Therefore, by Itô’s formula,

(4.7)

where we have performed the $k_2$ and $k_3$ integrations, and used that $|\rho _k| \leqslant 1$.

Recall that $\partial _{k'} \rho _{k'}^2$ is supported on frequencies $| n | \in (c_\rho k', C_\rho k')$. Hence, for any $\kappa > 0$,

$$\begin{aligned} \begin{aligned} (4.7)&\lesssim \frac{1}{N^3}\sum _{n_1 + n_2 + n_3 = n} \frac{1}{\langle n_2 \rangle ^2 \langle n_3 \rangle ^2} \int _0^K\Bigg ( \int _{k_1}^k \frac{\partial _{k'} \rho _{k'}^2(n)}{\langle n \rangle ^{2-\frac{\kappa }{2}}} dk' \Bigg )^2 \frac{\partial _{k_1}\rho _{k_1}^2(n_1)}{\langle n_1 \rangle ^2 \langle k_1 \rangle ^{\kappa }} dk_1 \\&\leqslant \frac{1}{N^3}\sum _{n_1 + n_2 + n_3 = n} \frac{1}{\langle n \rangle ^{4-\kappa }\langle n_2 \rangle ^2 \langle n_3 \rangle ^2} \int _0^k \frac{\partial _{k_1}\rho _{k_1}^2(n_1)}{\langle n_1 \rangle ^{2+\kappa }} dk_1 \\&\lesssim \frac{1}{N^3}\sum _{n_1 + n_2 + n_3 = n} \frac{1}{\langle n \rangle ^{4-\kappa }\langle n_1 \rangle ^{2+\kappa } \langle n_2 \rangle ^2 \langle n_3 \rangle ^2} \lesssim \frac{N^3}{\langle n \rangle ^4}, \end{aligned} \end{aligned}$$

(4.8)

where $\lesssim$ means $\leqslant$ up to a constant depending only on $\eta$, $c_\rho$ and $C_\rho$; the last inequality uses standard bounds on discrete convolutions contained in Lemma A.12; and we have used that the double convolution produces a volume factor of $N^6$. Note that, as said in Sect. 2, we omit the dependence on $c_\rho$ and $C_\rho$ in the final bound.

By Fubini’s theorem, Nelson’s hypercontractivity estimate [Nel73] (or the related Burkholder-Davis-Gundy inequality [RY13, Theorem 4.1]), and space-stationarity

(4.9)

where $\Delta _j$ is the j-th Littlewood-Paley block defined in Appendix A and we recall .

We overload notation and also write $\Delta _j$ to mean its corresponding Fourier multiplier. Then, by space-stationarity, for any $j \geqslant -1$,

(4.10)

Inserting (4.10) into (4.9) we obtain

which converges provided $s < \frac{1}{2}$, thus finishing the proof. $\quad \square$

4.2 The Boué–Dupuis formula

Fix an ultraviolet cutoff K. Recall that we are interested in Gaussian expectations of the form

$$\begin{aligned} {\mathbb {E}}_N e^{-{\mathcal {H}}(\phi _K)} \end{aligned}$$

where ${\mathcal {H}}(\phi _K) = {\mathcal {H}}_{\beta ,N,K}(\phi _K) + Q_K(\phi _K)$.

We may represent such expectations on $(\Omega , {\mathcal {A}}, ({\mathcal {A}}_k)_{k \geqslant 0}, {\mathbb {P}})$:

(4.11)

The key point is that the righthand side of (4.11) is written in terms of a measurable functional of Brownian motions. This allows us to exploit continuous time martingale techniques, crucially Girsanov’s theorem [RY13, Theorems 1.4 and 1.7], to reformulate (4.11) as a stochastic control problem.

Let ${\mathbb {H}}$ be the set of processes $v_\bullet$ that are ${\mathbb {P}}$-almost surely in $L^2({\mathbb {R}}_+;L^2({\mathbb {T}}_N))$ and progressively measurable with respect to $({\mathcal {A}}_k)_{k \geqslant 0}$. We call this the space of drifts. For any $v \in {\mathbb {H}}$, let $V_\bullet$ be the process defined by

$$\begin{aligned} V_k = \int _0^k {\mathcal {J}}_{k'} v_{k'} d{k'}. \end{aligned}$$

For our purposes, it is sufficient to consider the subspace of drifts ${\mathbb {H}}_K \subset {\mathbb {H}}$ consisting of $v \in {\mathbb {H}}$ such that $v_k = 0$ for $k > K$.

We also work with the subset of bounded drifts ${\mathbb {H}}_{b,K} \subset {\mathbb {H}}_K$, defined as follows: for every $M \in {\mathbb {N}}$, let ${\mathbb {H}}_{b,M,K}$ be the set of $v \in {\mathbb {H}}_K$ such that

$$\begin{aligned} \int _0^K \int _{{\mathbb {T}}_N}v_k^2 dx dk \leqslant M \end{aligned}$$

(4.12)

${\mathbb {P}}$-almost surely. Set ${\mathbb {H}}_{b,K} = \bigcup _{M \in {\mathbb {N}}} {\mathbb {H}}_{b,M,K}$.

The following proposition is the main tool of this section.

Proposition 4.7

Let $N \in {\mathbb {N}}$ and ${\mathcal {H}}:C^\infty ({\mathbb {T}}_N) \rightarrow {\mathbb {R}}$ be measurable and bounded. Then, for any $K > 0$,

(4.13)

where the infimum can be taken over v in ${\mathbb {H}}_K$ or ${\mathbb {H}}_{b,K}$.

Proof

(4.13) was first established by Boué and Dupuis [BD98], but we use the version in [BD19, Theorem 8.3], adapted to our setting. $\quad \square$

We cannot directly apply Proposition 4.7 for the case ${\mathcal {H}}= {\mathcal {H}}_{\beta ,N,K} + Q_K$ because it is not bounded. To circumvent this technicality, we introduce a total energy cutoff $E \in {\mathbb {N}}$. Since K is taken fixed, ${\mathcal {H}}_{\beta ,N,K} + Q_K$ is bounded from below. Hence, by dominated convergence

$$\begin{aligned} \lim _{E \rightarrow \infty }{\mathbb {E}}_N e^{-\big ({\mathcal {H}}_{\beta ,N,K}(\phi _K) + Q_K(\phi _K)\big ) \wedge E} = {\mathbb {E}}_N e^{- {\mathcal {H}}_{\beta ,N,K}(\phi _K) + Q_K(\phi _K)}. \end{aligned}$$

(4.14)

We apply Proposition 4.7 to ${\mathcal {H}}= \big ({\mathcal {H}}_{\beta ,N,K} + Q_K\big ) \wedge E$. For the lower bound on the corresponding variational problem, we establish estimates that are uniform over $v \in {\mathbb {H}}_{b,K}$. For the upper bound, we establish estimates for a specific choice of $v \in {\mathbb {H}}_K$ which is constructed via a fixed point argument. All estimates that we establish are independent of E. Hence, using (4.14) and the representation (4.11), they carry over to ${\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K) + Q_K(\phi _K)}$. We suppress mention of E unless absolutely necessary.

Remark 4.8

The assumption that ${\mathcal {H}}$ is bounded allows the infimum in (4.13) to be interchanged between ${\mathbb {H}}_K$ and ${\mathbb {H}}_{b,K}$. The use of ${\mathbb {H}}_{b,K}$ allows one to overcome subtle stochastic analysis issues that arise later on: specifically, justifying certain stochastic integrals appearing in Lemmas 5.15 and 5.17 are martingales and not just local martingales. See Lemma 5.14. The additional boundedness condition is important in the lower bound on the variational problem as the only other a priori information that we have on v there is that ${\mathbb {E}}\int _0^K\int _{{\mathbb {T}}_N}v_k^2 dx dk < \infty$, which alone is insufficient. On the other hand, the candidate optimiser for the upper bound is constructed in ${\mathbb {H}}_K$, but it has sufficient moments to guarantee the aforementioned stochastic integrals in Lemma 5.14 are martingales. See Lemma 5.22.

Remark 4.9

A version of the Boué–Dupuis formula for ${\mathcal {H}}$ measurable and satisfying certain integrability conditions is given in [Üst14, Theorem 7]. These integrability conditions are broad enough to cover the cases that we are interested in, and it is required in [BG19] to identify the Laplace transform of $\phi ^4_3$. However, it is not clear to us that the infimum in the corresponding variational formula can be taken over ${\mathbb {H}}_{b,K}$. Therefore, it seems that the stochastic analysis issues discussed in Remark 4.8 cannot be resolved directly using this version without requiring some post-processing (e.g. via a dominated convergence argument with a total energy cutoff as above).

4.2.1 Relationship with the Gibbs variational principle

Given a drift $v \in {\mathbb {H}}_K$, we define the measure ${\mathbb {Q}}$ whose Radon-Nikodym derivative with respect to ${\mathbb {P}}$ is given by the following stochastic exponential:

$$\begin{aligned} d{\mathbb {Q}}= e^{\int _0^Kv_k dX_k - \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^Kv_k^2 dk dx } d{\mathbb {P}}. \end{aligned}$$

(4.15)

Let ${\mathbb {H}}_{c,K}$ be the set of $v \in {\mathbb {H}}_K$ such that its associated measure defined in (4.15) is a probability measure, i.e. the expectation of the stochastic integral is 1. Then, by Girsanov’s theorem [RY13, Theorems 1.4 and 1.7 in Chapter VIII] it follows that the process $X_\bullet$ is a semi-martingale under ${\mathbb {Q}}$ with decomposition:

$$\begin{aligned} X_K = X_K^v + \int _0^K v_k dx \end{aligned}$$

where $X^v_\bullet$ is an $L^2$ cylindrical Brownian motion with respect to ${\mathbb {Q}}$. This induces the decomposition

(4.16)

where .

Lemma 4.10

Let $N \in {\mathbb {N}}$ and ${\mathcal {H}}:C^\infty ({\mathbb {T}}_N) \rightarrow {\mathbb {R}}$ be measurable and bounded from below. Then, for any $K > 0$,

(4.17)

where ${\mathbb {E}}_{\mathbb {Q}}$ denotes expectation with respect to ${\mathbb {Q}}$.

Proof

(4.17) is a well-known representation of the classical Gibbs variational principle [DE11, Proposition 4.5.1]. Indeed, one can verify that $R({\mathbb {Q}}\Vert {\mathbb {P}}) = {\mathbb {E}}_{{\mathbb {Q}}} \Big [ \int _0^\infty \int _{{\mathbb {T}}_N}v_k^2 dx dk\Big ]$, where $R({\mathbb {Q}}\Vert {\mathbb {P}}) = {\mathbb {E}}_{{\mathbb {Q}}} \log \frac{d{\mathbb {Q}}}{d{\mathbb {P}}}$ is the relative entropy of ${\mathbb {Q}}$ with respect to ${\mathbb {P}}$. A full proof in our setting is given in [GOTW18, Proposition 4.4]. $\quad \square$

Proposition 4.7 has several upshots over Lemma 4.10. The most important for us is that drifts can be taken over a Banach space, thus allowing candidate optimisers to be constructed using fixed point arguments via contraction mapping. In addition, the underlying probability space is fixed (i.e. with respect to the canonical measure ${\mathbb {P}}$), although this is a purely aesthetic advantage in our case. The cost of these upshots is that the minimum in (4.17) is replaced by an infimum in (4.13), and more rigid conditions on ${\mathcal {H}}$ are required. We refer to [BD19, Section 8.1.1] or [BG19, Remark 1] for further discussion.

With the connection with the Gibbs variational principle in mind, we call ${\mathcal {H}}(V_K)$ the drift (potential) energy and we call $\int _0^K\int _{{\mathbb {T}}_N}v_k^2 dx dk$ the drift entropy.

4.2.2 Regularity of the drift

In our analysis we use intermediate scales between 0 and K. As we explain in Sect. 5.1, this means that we require control over the process $V_\bullet$ in terms of the drift energy and drift entropy terms in (4.13).

The drift entropy allows a control of $V_\bullet$ in $L^2$-based topologies.

Lemma 4.11

For every $v \in L^2({\mathbb {R}}_+;L^2({\mathbb {T}}_N))$ and $K > 0$,

(4.18)

Proof

(4.18) is a straightforward consequence definition of ${\mathcal {J}}_k$, see [BG19, Lemma 2]. $\quad \square$

To control the homogeneity in our estimates, we also require bounds on $\Vert V_\bullet \Vert _{L^4}^4$. This is a problem: for our specific choices of ${\mathcal {H}}$, the drift energy allows a control in $L^4$-based topologies at the endpoint $V_K$. It is in general impossible to control the history of the path by the endpoint (for example, consider an oscillating process $V_\bullet$ with $V_K = 0$). We follow [BG19] to sidestep this issue.

Let ${\tilde{\rho }} \in C^\infty _c({\mathbb {R}}_+;{\mathbb {R}}_+)$ be non-increasing such that

$$\begin{aligned} {\tilde{\rho }}(x) = {\left\{ \begin{array}{ll} 1 &{} \quad |x| \in \Big [ 0, \frac{c_\rho }{2} \Big ] \\ 0 &{} \quad |x| \in \Big [ c_\rho , \infty \Big ) \end{array}\right. } \end{aligned}$$

and let ${\tilde{\rho }}_k(\cdot ) = {\tilde{\rho }}(\frac{\cdot }{k})$ for every $k>0$.

Define the process $V^\flat _\bullet$ by

$$\begin{aligned} V_k^\flat&= \frac{1}{N^3} \sum _{n} {\tilde{\rho }}_k(n) \Bigg (\int _0^k {\mathcal {J}}_{k'}(n) {\mathcal {F}}{v_{k'}}(n) dk'\Bigg )e^n. \end{aligned}$$

Note that ${\mathcal {F}}(V_k^\flat )(n) = {\mathcal {F}}(V_k)(n)$ if $|n| \leqslant \frac{c_\rho }{2}$. Thus, $V^\flat _\bullet$ and $V_\bullet$ have the same low frequency/large-scale behaviour (hence the notation).

The two processes differ on higher frequencies/small-scales. Indeed, as a Fourier multiplier, ${\tilde{\rho }}_k {\mathcal {J}}_k = 0$ for $k' > k$. Hence, for any $k \leqslant K$,

$$\begin{aligned} V_k^\flat = \frac{1}{N^3} \sum _{n} {\tilde{\rho }}_k(n) \Bigg (\int _0^K {\mathcal {J}}_{k'}(n) {\mathcal {F}}{v_{k'}}(n) dk'\Bigg )e^n = {\tilde{\rho }}_k V_K. \end{aligned}$$

This is sufficient for our purposes because ${\tilde{\rho }}_k$ is an $L^p$ multiplier for $p \in (1,\infty )$, and hence the associated operator is $L^p$ bounded for $p \in (1,\infty )$.

Lemma 4.12

For any $p \in (1,\infty )$, there exists $C=C(p,\eta )>0$ such that, for every $v \in L^2({\mathbb {R}}_+;L^2({\mathbb {T}}_N))$,

$$\begin{aligned} \sup _{0 \leqslant k \leqslant K}\Vert V_k^\flat \Vert _{L^p} \leqslant C\Vert V_K\Vert _{L^p}. \end{aligned}$$

(4.19)

Moreover, for any $s,s' \in {\mathbb {R}}$, $p \in (1,\infty )$, $q \in [1,\infty ]$, there exists $C=C(s,s',p,q,\eta )$ such that, for every $v \in L^2({\mathbb {R}}_+;L^2({\mathbb {T}}_N))$,

$$\begin{aligned} \sup _{0 \leqslant k \leqslant K}\Vert \partial _k V_k^\flat \Vert _{B^{s'}_{p,q}} \leqslant C \frac{\Vert V_K\Vert _{B^s_{p,q}}}{\langle k \rangle ^{1+s-s'}}. \end{aligned}$$

(4.20)

Proof

(4.19) and (4.20) are a consequence of the preceding discussion together with the observation that $\partial _k V_k^\flat$ is supported on an annulus in Fourier space and, subsequently, applying Bernstein’s inequality (A.6). See [BG19, Lemma 20]. $\quad \square$

5 Estimates on Q-Random Variables

The main results of this section are upper bounds on expectations of certain random variables, derived from $Q_1,Q_2,$ and $Q_3$ defined in (3.2), that are uniform in $\beta$ and extensive in $N^3$.

Proposition 5.1

For every $a_0 > 0$, there exist $\beta _0 = \beta _0(a_0,\eta ) \geqslant 1$ and $C_Q = C_Q(a_0, \beta _0, \eta )>0$ such that the following estimates hold: for all $\beta > \beta _0$ and $a \in {\mathbb {R}}$ satisfying $|a| \leqslant a_0$,

In addition,

where B is any set of unordered pairs of nearest-neighbour blocks that partitions ${{\mathbb {B}}_N}$.

Proof

See Sect. 5.9. $\quad \square$

Proposition 5.1 is used in Sect. 6.3, together with the chessboard estimates of Proposition 6.5, to prove Proposition 3.6. Indeed, chessboard estimates allow us to obtain estimates on expectations of random variables, derived from the $Q_i$, that are extensive in their support from estimates that are extensive in $N^3$. Note that the latter are significantly easier to obtain than the former since these random variables may be supported on arbitrary unions of blocks.

Remark 5.2

For the remainder of this section, we assume $\eta < \frac{1}{392 C_P}$ where $C_P$ is the Poincaré constant on unit blocks (see Proposition A.11). This is for convenience in the analysis of Sects. 5.8.1 and 5.9 (see also Lemma 5.21). Whilst this may appear to fix the specific choice of renormalisation constants $\delta m^2$, we can always shift into this regime by absorbing a finite part of $\delta m^2$ into ${\mathcal {V}}_\beta$.

Most of the difficulties in the proof of Proposition 5.1 are contained in obtaining the following upper and lower bounds on the free energy $-\log {\mathscr {Z}}_{\beta ,N}$ that are uniform in $\beta$.

Proposition 5.3

There exists $C=C(\eta )>0$ such that, for all $\beta \geqslant 1$,

$$\begin{aligned} \liminf _{K \rightarrow \infty } -\frac{1}{N^3}\log {\mathscr {Z}}_{\beta ,N,K} \geqslant - C \end{aligned}$$

(5.1)

and

$$\begin{aligned} \limsup _{K \rightarrow \infty } -\frac{1}{N^3} \log {\mathscr {Z}}_{\beta ,N,K} \leqslant C. \end{aligned}$$

(5.2)

Proof

See Sects. 5.8.1 and 5.8.2 for a proof of (5.1) and (5.2), respectively. These proofs rely on Sects. 5.2–5.7, and the overall strategy is sketched in Sect. 5.1. $\quad \square$

Remark 5.4

In [BG19] estimates on $-\log {\mathscr {Z}}_{\beta ,N,K}$ are obtained that are uniform in $K > 0$ and extensive in $N^3$. However, one can show that these estimates are $O(\beta )$ as $\beta \rightarrow \infty$. This is insufficient for our purposes (compare with the uniform in $\beta$ estimates required to prove Proposition 3.2).

5.1 Strategy to prove Proposition 5.3

The lower bound on $-\log {\mathscr {Z}}_{\beta ,N,K}$, given by (5.1), is the harder bound to establish in Proposition 5.3. Our approach builds on the analysis of [BG19] by incorporating a low temperature expansion inspired by [GJS76a, GJS76b]. This is explained in more detail in Sect. 5.1.1.

On the other hand, we establish the upper bound on $-\log {\mathscr {Z}}_{\beta ,N,K}$, given by (5.2), by a more straightforward modification of the analysis in [BG19]. See Sect. 5.8.2.

We now motivate our approach to establishing (5.1) by first isolating the the difficulty in obtaining $\beta$-independent bounds when using [BG19] straight out of the box. The starting point is to apply Proposition 4.7 with ${\mathcal {H}}= {\mathcal {H}}_{\beta ,N,K}$, together with a total energy cutoff that we refrain from making explicit (see Remark 4.8 and the discussion that precedes it), to represent $-\log {\mathscr {Z}}_{\beta ,N,K}$ as a stochastic control problem.

For every $v \in {\mathbb {H}}_{b,K}$, define

Ultraviolet divergences occur in the expansion of since the integrals and appear and cannot be bounded uniformly in K:

• For the first integral, there are difficulties in even interpreting as a random distribution in the limit $K \rightarrow \infty$. Indeed, the variance of tested against a smooth function diverges as the cutoff is removed.

• On the other hand, one can show that does converge as $K \rightarrow \infty$ to a random distribution of Besov-Hölder regularity $-1-\kappa$ for any $\kappa > 0$ (see Proposition 4.4). However, this regularity is insufficient to obtain bounds on the second integral uniform on K . Indeed, $V_K$ can be bounded in at most $H^1$ uniformly in K (see Lemma 4.11), and hence we cannot test against $V_K$ (or $V_K^2$) in the limit $K \rightarrow \infty$.

This is where the need for renormalisation beyond Wick ordering appears.

To implement this, we follow [BG19] and postulate that the small-scale behaviour of the drift v is governed by explicit renormalised polynomials of through the change of variables:

(5.3)

where the remainder term $r=r(v)$ is defined by (5.3). Since $v \in {\mathbb {H}}_K \supset {\mathbb {H}}_{b,K}$, we have that $r \in {\mathbb {H}}_K$ and, hence, has finite drift entropy; however, note that $r \not \in {\mathbb {H}}_{b,K}$. The optimisation problem is then changed from optimising over $v \in {\mathbb {H}}_{b,K}$ to optimising over $r(v) \in {\mathbb {H}}_K$.

The change of variables (5.3) means that the drift entropy of any v now contains terms that are divergent as $K \rightarrow \infty$. One uses Itô’s formula to decompose the divergent integrals identified above into intermediate scales, and then uses these divergent terms in the drift entropy to mostly cancel them. Using the renormalisation counterterms beyond Wick ordering (i.e. the terms involving $\gamma _K$ and $\delta _K$), the remaining divergences can be written in terms of well-defined integrals involving the diagrams defined in Sect. 4.1.1.

One can then establish that, for every $\varepsilon >0$, there exists $C=C(\varepsilon ,\beta ,\eta )>0$ such that, for every $v \in {\mathbb {H}}_{b,K}$,

$$\begin{aligned} {\mathbb {E}}\Psi _K(v) \geqslant -CN^3 + (1-\varepsilon ){\mathbb {E}}[G_K(v)] \end{aligned}$$

(5.4)

where

$$\begin{aligned} G_K(v) = \frac{1}{\beta }\int _{{\mathbb {T}}_N}V_K^4 dx + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}r_k^2 dk dx \geqslant 0. \end{aligned}$$

The quadratic term in $G_K(v)$ allows one to control the $H^{\frac{1}{2} -\kappa }$ norm of $V_K$ for any $\kappa > 0$, uniformly in K (see Proposition 5.9). These derivatives on $V_K$ appear when analysing terms in $\Psi _K(v)$ involving Wick powers of tested against (powers of) $V_K$. However, some of these integrals have quadratic or cubic dependence on the drift, thus the quadratic term in $G_K(v)$ is insufficient to control the homogeneity in these estimates; instead, this achieved by using the quartic term in $G_K(v)$. Note that the good sign of the quartic term in the ${\mathcal {H}}_{\beta ,N,K}$ ensures that $G_K(v)$ is indeed non-negative.

Using the representation (4.11) on ${\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K)}$ and applying Proposition 4.7, one obtains $-\log {\mathscr {Z}}_{\beta ,N,K} \geqslant - CN^3$ from (5.4) and the positivity of $G_K(v)$.

As pointed out in Remark 5.4, this argument gives $C=O(\beta )$ for $\beta$ large and this is insufficient for our purposes. The suboptimality in $\beta$-dependence comes from the treatment of the integral

$$\begin{aligned} \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K) - \frac{\eta }{2} V_K^2 dx \end{aligned}$$

(5.5)

in . The choice of $G_K(v)$ in the preceding discussion is not appropriate in light of (5.5) since the term $\int _{{\mathbb {T}}_N}V_K^4 dx$ destroys the structure of the non-convex potential $\int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K) dx$. On the other hand, replacing $\frac{1}{\beta }\int _{{\mathbb {T}}_N}V_K^4 dx$ with the whole integral (5.5) in $G_K(v)$ does not work. This is because (5.5) does not admit a $\beta$-independent lower bound.

5.1.1 Fixing $\beta$ dependence via a low temperature expansion

We expand (5.5) as two terms

$$\begin{aligned} (5.5) = \int _{{\mathbb {T}}_N}\frac{1}{2} {\mathcal {V}}_\beta (V_K) dx + \int _{{\mathbb {T}}_N}\frac{1}{2} {\mathcal {V}}_\beta (V_K) - \frac{\eta }{2} V_K^2 dx. \end{aligned}$$

(5.6)

The first integral in (5.6) is non-negative so we use it as a stability/good term for the deterministic analysis, i.e. replacing $G_K(v)$ by

$$\begin{aligned} \int _{{\mathbb {T}}_N}\frac{1}{2} {\mathcal {V}}_\beta (V_K) dx + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}r_k^2 dx dk. \end{aligned}$$

(5.7)

This requires a comparison of $L^p$ norms of $V_K$ for $p \leqslant 4$ on the one hand, and $\int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K) dx$ on the other. Due to the non-convexity of ${\mathcal {V}}_\beta$, this produces factors of $\beta$; these have to be beaten by the good (i.e. negative) powers of $\beta$ appearing in . We state the required bounds in the following lemma.

Lemma 5.5

For any $p\in [1,4]$, there exists $C=C(p) > 0$ such that, for all $a \in {\mathbb {R}}$,

$$\begin{aligned} |a|^p \leqslant C ({\sqrt{\beta }})^\frac{p}{2} {\mathcal {V}}_\beta (a)^\frac{p}{4} + C ({\sqrt{\beta }})^p. \end{aligned}$$

(5.8)

Hence, for any $f \in C^\infty ({\mathbb {T}}_N)$,

(5.9)

where we recall .

Proof

(5.8) follows from a straightforward computation. (5.9) follows from using (5.8) and Jensen’s inequality. $\quad \square$

The difficulty lies in bounding the second integral in (5.6) uniformly in $\beta$. In 2D an analogous problem was overcome in [GJS76a, GJS76b] in the context of a low temperature expansion for $\Phi ^4_2$. Those techniques rely crucially on the logarithmic ultraviolet divergences in 2D, and the mutual absolute continuity between $\Phi ^4_2$ and its underlying Gaussian measure. Thus, they do not extend to 3D. However, we use the underlying strategy of that low temperature expansion in our approach.

We write ${\mathscr {Z}}_{\beta ,N,K}$ as a sum of $2^{N^3}$ terms, where each term is a modified partition function that enforces the block averaged field to be either positive or negative on blocks. For each term in the expansion, we change variables and shift the field on blocks to $\pm {\sqrt{\beta }}$ so that the new mean of the field is small. We then apply Proposition 4.7 to each of these $2^{N^3}$ terms.

We separate the scales in the variational problem by coarse-graining the resulting Hamiltonian. Large scales are captured by an effective Hamiltonian, which is of a similar form to the second integral in (5.6). We treat this using methods inspired by [GJS76b, Theorem 3.1.1]: the expansion and translation allow us to obtain a $\beta$-independent bound on the effective Hamiltonian with an error term that depends only on the difference between the field and its block averages (the fluctuation field). The fluctuation field can be treated using the massless part of the underlying Gaussian measure (compare with [GJS76b, Proposition 2.3.2]).

The remainder term contains all the small-scale/ultraviolet divergences and we renormalise them using the pathwise approach of [BG19] explained above. Patching the scales together requires uniform in $\beta$ estimates on the error terms from the renormalisation procedure using an analogue of the stability term (5.7) that incorporates the translation, and Lemma 5.5.

5.2 Expansion and translation by macroscopic phase profiles

Let $\chi _+,\chi _-:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be defined as

$$\begin{aligned} \chi _+(a) = \frac{1}{\sqrt{\pi }} \int _{-a}^\infty e^{-c^2} dc, \quad \chi _-(a) = \chi _+(-a). \end{aligned}$$

They satisfy

$$\begin{aligned} \chi _+(a) + \chi _-(a) = 1 \end{aligned}$$

and hence

for any .

For any $K>0$, we expand

(5.10)

where we recall .

We fix $\sigma$ in what follows and sometimes suppress it from notation. Let $h = {\sqrt{\beta }}\sigma$. We then have

We translate the Gaussian fields so that their new mean is approximately h. The translation we use is related to the classical magnetism, or response to the external field $\eta h$, used in the 2D setting [GJS76a] and given by $\eta (-\Delta +\eta )^{-1} h$.

Lemma 5.6

For every $K>0$, let $h_K = \rho _K h$. Define ${\tilde{g}}_K = \eta (-\Delta + \eta )^{-1} h_K$ and $g_K = \rho _K {\tilde{g}}_K$. Then, there exists $C=C(\eta )$ such that

$$\begin{aligned} | g_K |_{\infty }, | \nabla g_K |_{\infty } \leqslant C{\sqrt{\beta }}\end{aligned}$$

(5.11)

where $| \cdot |_{\infty }$ denotes the supremum norm. Moreover,

$$\begin{aligned} \int _{{\mathbb {T}}_N}|\nabla g_K|^2 dx \leqslant \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx. \end{aligned}$$

(5.12)

Finally, let

$$\begin{aligned} g_k^\flat = \sum _{n \in (N^{-1}{\mathbb {Z}})^3} \frac{1}{N^3} {\tilde{\rho }}_k \int _0^k {\mathcal {J}}_{k'}(n) {\mathcal {F}}g (n) dk \end{aligned}$$

where ${\tilde{\rho }}_k$ is as in Sect. 4.2.2. Then, for any $s,s' \in {\mathbb {R}}$, $p \in (1,\infty )$ and $q \in [1,\infty ]$, there exists $C_1=C_1(\eta ,s,p,q)$ and $C_2=C_2(\eta ,s,s',p,q)$ such that

$$\begin{aligned} \Vert g_k^\flat \Vert _{B^s_{p,q}} \leqslant C_1\Vert g_K \Vert _{B^s_{p,q}} \end{aligned}$$

(5.13)

and

$$\begin{aligned} \Vert \partial _k g_k^\flat \Vert _{B^{s'}_{p,q}} \leqslant C_2\frac{1}{\langle k \rangle ^{1+s-s'}} \Vert g_K \Vert _{B^s_{p,q}}. \end{aligned}$$

(5.14)

Proof

The estimate (5.11) follows from the fact that $\eta (-\Delta + \eta )^{-1}$ and $\nabla \eta (-\Delta + \eta )^{-1}$ are $L^\infty$ bounded operators. This is because the ($\eta$-dependent) Bessel potential and its first derivatives are absolutely integrable on ${\mathbb {R}}^3$. Hence, by applying Young’s inequality for convolutions one obtains the $L^\infty$ boundedness. The uniformity of the estimate over $\sigma$ follows from $\Vert \sigma \Vert _{L^\infty }=1$ for every $\sigma \in {{\mathbb {B}}_N}$. The other estimates follow from standard results about smooth multipliers, the observation that $g_k^\flat = {\tilde{\rho }}_k g_K$ for any $K \geqslant k$, and Lemma 4.12. $\quad \square$

Remark 5.7

Note that $g_K$ is given by the covariance operator of $\mu _N$ applied to $\eta h$. Moreover, note that $g_K \ne {\tilde{g}}_K$ since $\rho _K^2 \ne \rho _K$, i.e. the Fourier cutoff is not sharp.

By the Cameron-Martin theorem the density of $\mu _N$ under the translation $\phi = \psi + {\tilde{g}}_K$ transforms as

$$\begin{aligned} d\mu _N(\psi +{\tilde{g}}_K) = \exp \Big ( - \int _{{\mathbb {T}}_N}\frac{1}{2} {\tilde{g}}_K (-\Delta + \eta ) {\tilde{g}}_K + \psi (-\Delta + \eta ){\tilde{g}}_K dx \Big ) d\mu _N(\psi ). \end{aligned}$$

Hence,

$$\begin{aligned} {\mathscr {Z}}_{\beta ,N,K}^\sigma = {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^\sigma (\psi _K) - F_{\beta ,N,K}^\sigma }(\psi ) \end{aligned}$$

where

and

$$\begin{aligned} F_{\beta ,N,K}^\sigma (\psi ) = \int _{{\mathbb {T}}_N}-\eta (\psi _K + g_K) h + \frac{\eta }{2} h^2 + \frac{1}{2} {\tilde{g}}_K (-\Delta + \eta ) {\tilde{g}}_K + \psi (-\Delta + \eta ){\tilde{g}}_K dx. \end{aligned}$$

By integration by parts, the self-adjointness of $\rho _K$, and the definition of ${\tilde{g}}_K$

$$\begin{aligned} \begin{aligned} F_{\beta ,N,K}^\sigma (\psi )&= \int _{{\mathbb {T}}_N}- \eta (\psi + {\tilde{g}}_K) h_K + \frac{\eta }{2} h^2 + \frac{1}{2} |\nabla {\tilde{g}}_K|^2 + \frac{\eta }{2} ({\tilde{g}}_K)^2 + \eta \psi h_K dx \\&= \int _{{\mathbb {T}}_N}\frac{\eta }{2} ({\tilde{g}}_K - h_K)^2 + \frac{\eta }{2} (1-\rho _K^2) h^2 + \frac{1}{2} |\nabla {\tilde{g}}_K|^2 dx. \end{aligned} \end{aligned}$$

(5.15)

Thus, $F^\sigma _{\beta ,N,K}(\psi )$ is independent of $\psi$ and non-negative.

Remark 5.8

Let $g = \eta (-\Delta + \eta )^{-1} h$. Then,

$$\begin{aligned} \lim _{K \rightarrow \infty } F^\sigma _{\beta ,N,K} = \int _{{\mathbb {T}}_N}\frac{\eta }{2} (g-h)^2 + \frac{1}{2} |\nabla g|^2 dx. \end{aligned}$$

(5.16)

The second integrand on the righthand side of (5.16) penalises the discontinuities of $\sigma$. Indeed, $e^{-\int _{{\mathbb {T}}_N}\frac{1}{2} |\nabla g|^2 dx}$ is approximately equal to $e^{-C{\sqrt{\beta }}|\partial \sigma |}$, where $\partial \sigma$ denotes the surfaces of discontinuity of $\sigma$, $|\partial \sigma |$ denotes the area of these surfaces, and $C>0$ is an inessential constant. Thus, for $\beta$ sufficiently large, ${\mathscr {Z}}_{\beta ,N}$ is approximately equal to

$$\begin{aligned} e^{- C \sqrt{\beta }|\partial \sigma |} \times O(1) = \prod _{\Gamma _i \in \sigma } e^{-C {\sqrt{\beta }}|\Gamma _i|} \times O(1) \end{aligned}$$

where $\Gamma _i$ are the connected components of $\partial \sigma$ (called contours). It would be interesting to further develop this contour representation for ${\nu _{\beta ,N}}$ (compare with the 2D expansions of [GJS76a, GJS76b]).

5.3 Coarse-graining of the Hamiltonian

We apply Proposition 4.7 to . For every $v \in {\mathbb {H}}_{b,K}$, define

(5.17)

Let , where . We split the Hamiltonian as

(5.18)

where

is an effective Hamiltonian introduced to capture macroscopic scales of the system. The quantity ${\mathcal {R}}_K$ is then determined by (5.18) and is explicitly given by

All analysis/cancellation of ultraviolet divergences occurs within the sum of ${\mathcal {R}}_K$ and the drift entropy, see (5.27). Finally, the last term in (5.18) is a stability term which is key for our non-perturbative analysis, namely it allows us to obtain estimates that are uniform in the drift.

The key point is that we coarse-grain the field by block averaging , the most singular term. This allows us to preserve the structure of the low temperature potential ${\mathcal {V}}_\beta$ on macroscopic scales (captured in ${\mathcal {H}}_K^\mathrm {eff}(Z_K)$), which is crucial to obtaining estimates independent of $\beta$ on the free energy.

5.4 Killing divergences

5.4.1 Changing drift variables

For any $v \in {\mathbb {H}}_{b,K}$, define $r=r(v) \in {\mathbb {H}}_{K}$ by

(5.19)

In our analysis it is convenient to use an intermediate change of variables for the drift. Define $u=u(v) \in {\mathbb {H}}_K$ by

(5.20)

Inserting (5.19) and (5.20) into the definition of the integrated drift, $V_k = \int _0^k {\mathcal {J}}_{k'} v_{k'} dk'$, we obtain

(5.21)

where $R_k = \int _0^k {\mathcal {J}}_{k'} r_{k'} dk'$ and $U_k = \int _0^k {\mathcal {J}}_{k'} u_{k'} dk'$.

The following proposition contains useful estimates estimates on $U_K$ and $V_K$.

Proposition 5.9

For any $\varepsilon > 0$ and $\kappa > 0$ sufficiently small, there exists $C=C(\varepsilon ,\kappa ,\eta ) > 0$ such that, for all $\beta > 1$,

(5.22)

(5.23)

where we recall ; and $N^\Xi _K$ is a positive random variable on $\Omega$ that is ${\mathbb {P}}$-almost surely given by a finite linear combination of powers of (finite integrability) Besov and Lebesgue norms of the diagrams on the interval [0, K].

Proof

See Sect. 5.6.1. $\quad \square$

Remark 5.10

As a consequence of Proposition 4.4, the random variable $N^\Xi _K$ satisfies the following estimate: there exists $C = C(\eta )>0$ such that

$$\begin{aligned} {\mathbb {E}}N^\Xi _K \leqslant C N^3. \end{aligned}$$

(5.24)

In the following we denote by $N^\Xi _K$ any positive random variable on $\Omega$ that satisfies (5.24). In practice it is always ${\mathbb {P}}$-almost surely given by a finite linear combination of powers of (finite integrability) Besov norms of the diagrams in $\Xi$ on [0, K]. Note that $N^\Xi _K$ includes constants of the form $C=C(\eta )>0$.

5.4.2 The main small-scale estimates

In the following we write $\approx$ to mean equal up to a term with expectation 0 under ${\mathbb {P}}$.

Proposition 5.11

Let $\beta > 0$. For every $K>0$, define

(5.25)

where is defined in (4.4), and

(5.26)

Then, for every $v \in {\mathbb {H}}_{b,K}$,

$$\begin{aligned} {\mathcal {R}}_K + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}v_k^2 dk dx \approx \sum _{i=1}^4 {\mathcal {R}}^i_K + \frac{1}{2} \int _0^K \int _{{\mathbb {T}}_N} r_k^2 dx dk \end{aligned}$$

(5.27)

where

Moreover, the following estimate holds: for any $\varepsilon >0$, there exists $C=C(\varepsilon , \eta )>0$ such that, for all $\beta > 1$,

$$\begin{aligned} \begin{aligned} \max _{i=1,\dots ,4}\Big | {\mathcal {R}}^i_K \Big | \leqslant CN^\Xi _K + \varepsilon \Big ( \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx + \frac{1}{2} \int _0^K \int _{{\mathbb {T}}_N} r_k^2 dx dk \Big ) \end{aligned} \end{aligned}$$

(5.28)

where $N^\Xi _K$ is as in Remark 5.10.

Proof

We establish (5.27) in Sect. 5.5 by arguing as in [BG19, Lemma 5]. The remainder estimates (5.28) are then established in Sect. 5.6. $\quad \square$

Remark 5.12

The terms ${\mathcal {R}}^i_K$ are ordered according to their occurrence when isolating divergences for the readers’ convenience. For the estimates, however, it is more convenient to order them according to difficulty. Define

$$\begin{aligned} {\mathcal {R}}^a_K&= {\mathcal {R}}^{a,1}_K + {\mathcal {R}}^{a,2}_K + {\mathcal {R}}^{a,3}_K \end{aligned}$$

where

and let ${\mathcal {R}}^b_K = \sum _{i=1}^4 {\mathcal {R}}^i_K - {\mathcal {R}}^a_K$.

${\mathcal {R}}^a_K$ contains the most difficult terms to bound, either due to analytic considerations or $\beta$-dependence; ${\mathcal {R}}^b_K$ contains the terms that follow almost immediately from [BG19, Lemmas 18-23].

Remark 5.13

The products and appearing above are classically ill-defined in the limit $K \rightarrow \infty$. However, (probabilistic) estimates on the resonant product uniform in K are obtained in Proposition 4.4. Hence, the first product can be analysed using a paraproduct decompositions (A.7). The second product is less straightforward and requires a double paraproduct decomposition (see [BG19, Lemma 21 and Proposition 6] and [CC18, Proposition 2.22]).

5.5 Proof of (5.27): Isolating and cancelling divergences

Using that and all have expectation zero,

Hence, by reordering terms,

(5.29)

Ignoring the renormalisation counterterms (i.e. those involving $\gamma _K$ and $\delta _K$), the divergences in (5.29) are contained in the integrals and . In order to kill these divergences, we use changes of variables in the drift entropy to mostly cancel them; the remaining divergences are killed by the renormalisation counterterms. We renormalise the leading order divergences, ie. those polynomial in K, in Sect. 5.5.1. The divergences that are logarithmic in K are renormalised in Sect. 5.5.2.

In order to use the drift entropy to cancel divergences, we decompose certain (spatial) integrals across ultraviolet scales $k \in [0,K]$ using Itô’s formula. Error terms are produced that are stochastic integrals with respect to martingales (specifically, with respect to and ). The following lemma allows us to argue that these stochastic integrals are $\approx 0$.

Lemma 5.14

For any $v \in {\mathbb {H}}_{b,K}$, the stochastic integrals

(5.30)

and

(5.31)

are martingales. We recall that, above, $\Delta _i$ denotes the i-th Littlewood-Paley block.

Proof

In this proof, for any continuous local martingale $Z_\bullet$, we write for the corresponding quadratic variation process. Moreover, for any Z-adapted process $Y_\bullet$, we write $\int _0^KY_k \cdot dZ_k$ to denote the stochastic integral $\int _{{\mathbb {T}}_N}\int _0^KY_k dZ_k dx$.

We begin with two observations: first, let $v \in {\mathbb {H}}_{b,M,K}$ for some $M > 0$, i.e. those $v \in {\mathbb {H}}_{b,K}$ satisfying (4.18). Then, by Sobolev embedding, there exists $C=C(M,N,K,\eta )>0$ such that

$$\begin{aligned} \sup _{0 \leqslant k \leqslant K} \Vert V_k \Vert _{L^6}^6 \leqslant C \end{aligned}$$

${\mathbb {P}}$-almost surely.

Second, recalling the iterated integral representation of the Wick powers and (see e.g. (4.3)), one can show and . Thus, we can write the stochastic integrals (5.30) and (5.31) in terms of stochastic integrals with respect to . It suffices to show that their quadratic variations are finite in expectation.

Using that and by Young’s inequality,

Hence, (5.30) is a martingale.

Now consider (5.31). By (5.21),

where

Arguing as for (5.30), one can show .

By Young’s inequality and using that Littlewood-Paley blocks and the $\flat$ operator are $L^p$ multipliers, we have

thus establishing that (5.31) is a martingale. $\quad \square$

5.5.1 Energy renormalisation

In the next lemma, we cancel the leading order divergence using the change of variables (5.20) in the drift entropy. The error term does not depend on the drift and is divergent in expectation (as $K \rightarrow \infty$); it is cancelled by one part of the energy renormalisation $\delta _K$ (see (5.26)).

Lemma 5.15

Proof

By Itô’s formula, Lemma 5.14, and the self-adjointness of ${\mathcal {J}}_k$,

Hence, by (5.20),

$\square$

As a consequence of (5.20), the remaining (non-counterterm) integrals in (5.29) acquire additional divergences that are independent of the drift. We isolate them in the next lemma; they are also renormalised by parts of the energy renormalisation (see (5.26)).

Lemma 5.16

(5.32)

and

(5.33)

Proof

By (5.21),

(5.34)

Above we have used Wick’s theorem and the fact that is Wick ordered to conclude .

Similarly, . Hence, by (5.21)

(5.35)

Combining (5.34) and (5.35) establishes (5.32).

By (5.21),

where we have used that and, by Wick’s theorem, . This establishes (5.33). $\quad \square$

The divergences encountered in Lemmas 5.15 and 5.16 that are independent of the drift are killed by the energy renormalisation $\delta _K$ since, by definition,

(5.36)

5.5.2 Mass renormalisation

The integrals on the righthand side of (5.33) that involve the drift cannot be bounded uniformly as $K \rightarrow \infty$. We isolate divergences using a paraproduct decomposition and expand the drift entropy using (5.19) to mostly cancel them. This is done in Lemma 5.17. The remaining divergences are then killed in Lemma 5.18 using the mass renormalisation.

Lemma 5.17

(5.37)

Proof

We write

Thus, using (5.21) and a paraproduct decomposition on the most singular products,

(5.38)

All except the first two integrals are absorbed into ${\mathcal {R}}_K^3$.

For the first integral, we use the (drift-dependent) change of variables (5.19) in the drift entropy of u to mostly cancel the divergence. Due to the paraproduct term, using Itô’s formula to decompose into scales requires us to control $V_k + g_k$ for $k < K$. In order to be able to do this, we replace $V_K + g_K$ by $V_K^\flat + g_K^\flat$ first. Then, applying Itô’s formula, Lemma 5.14, and using the self-adjointness of ${\mathcal {J}}_k$,

(5.39)

From (5.19) and (5.20)

(5.40)

Combining (5.38), (5.39), and (5.40) yields (5.37). $\quad \square$

We now cancel the divergences in the last two terms of (5.37) using the mass renormalisation.

Lemma 5.18

(5.41)

Proof

By the definition of (see Sect. 4.1.1),

(5.42)

where we have used that, by Wick’s theorem, .

To renormalise the second integral in (5.41), we need to rewrite the remaining counterterm in terms of $V_K^\flat$:

$$\begin{aligned} \begin{aligned} -&\int _{{\mathbb {T}}_N}\frac{\gamma _K}{\beta ^2} (V_K+g_K)^2 dx \\&= -\int _{{\mathbb {T}}_N}\frac{\gamma _K}{\beta ^2} (V_K^\flat +g_K^\flat )^2 + 2(V_K^\flat +g_K^\flat )(V_K + g_K - V_K^\flat - g_K^\flat ) \\&\quad \quad \quad + (V_K + g_K -V_K^\flat - g_K^\flat )^2 dx. \end{aligned} \end{aligned}$$

(5.43)

Using Itô’s formula on the first integral of the right hand side of (5.43),

$$\begin{aligned} -&\int _{{\mathbb {T}}_N}\frac{\gamma _K}{\beta ^2} (V_K^\flat +g_K^\flat )^2 dx \\&= - \int _{{\mathbb {T}}_N}\int _0^K\frac{\partial _k{\gamma }_k}{\beta ^2} (V_k^\flat +g_k^\flat )^2 + \frac{2\gamma _k}{\beta ^2}( \partial _k V_k^\flat + \partial _k g_k^\flat )(V_k^\flat +g_k^\flat ) dk dx. \end{aligned}$$

By the definition of (see Sect. 4.1.1),

(5.44)

Hence, combining (5.42), (5.43), and (5.44) establishes (5.41). $\quad \square$

Proof of (5.27)

Lemmas 5.15, 5.16, 5.17, and 5.18 , together with (5.36), establish (5.27). $\quad \square$

5.6 Proof of (5.28): Estimates on remainder terms

Recall the definition of ${\mathcal {R}}^a_K$, which contains the most difficult terms to bound, and ${\mathcal {R}}^b_K$ from Remark 5.12.

Proposition 5.19

For any $\varepsilon > 0$, there exists $C=C(\varepsilon ,\eta )>0$ such that, for all $\beta > 1$,

$$\begin{aligned} |{\mathcal {R}}^{a,1}_K|&\leqslant C N^\Xi _K + \varepsilon \Big ( \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}r_k^2 dx dk \Big ) \end{aligned}$$

(5.45)

$$\begin{aligned} |{\mathcal {R}}^{a,2}_K|&\leqslant C N^\Xi _K + \varepsilon \Big ( \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}r_k^2 dx dk \Big ) \end{aligned}$$

(5.46)

$$\begin{aligned} |{\mathcal {R}}^{a,3}_K|&\leqslant C N^\Xi _K + \varepsilon \Big ( \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}r_k^2 dx dk \Big ). \end{aligned}$$

(5.47)

Proof

The estimates (5.45), (5.46), and (5.47) are established in Sects. 5.6.2, 5.6.3, and 5.6.4 respectively. (5.46) and (5.47) are established by a relatively straightforward combination of techniques in [BG19, Lemmas 18-23] together with Lemmas 5.5 and 5.6 . On the other hand, the terms with cubic dependence in the drift (5.45) require a slightly more involved analysis.

Note that, since our norms on functions/distributions were defined using instead of dx to track N dependence, in the proof we rewrite the integrals above in terms of by dividing both sides by $N^3$. $\quad \square$

Proposition 5.20

For any $\varepsilon > 0$, there exists $C=C(\varepsilon ,\eta )>0$ such that, for all $\beta > 1$,

$$\begin{aligned} |{\mathcal {R}}^b_K| \leqslant C N^\Xi _K + \varepsilon \Big ( \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx + \frac{1}{2} \int _0^K\int _{{\mathbb {T}}_N}r_k^2 dx dk \Big ). \end{aligned}$$

Proof

Follows from a direct combination of arguments in [BG19, Lemmas 18-23] with Lemmas 5.5 and 5.6 . We omit it. $\quad \square$

Proof of (5.28)

Since $\sum _{i=1}^4 {\mathcal {R}}_K^i = {\mathcal {R}}^a_K + {\mathcal {R}}^b_K$, Propositions 5.19 and 5.20 establish (5.28). $\quad \square$

The proofs of Propositions 5.19 and 5.20 rely heavily on bounds on the drift established in Proposition 5.9, so we prove this first in the next subsection. Throughout the remainder of this section, we use the notation $a \lesssim b$ to mean $a \leqslant Cb$ for some $C=C(\varepsilon ,\eta )$, and we also allow for this constant to depend on other inessential parameters (i.e. not $\beta$, N, or K).

5.6.1 Proof of Proposition 5.9

First, note that (5.23) is a direct consequence of (5.22) along with (5.21) and bounds contained in Proposition 4.4.

We now prove (5.22). Fix any $k' \in [0,K]$. As a consequence of (5.21),

(5.48)

By Minkowski’s integral inequality, Bernstein’s inequality (A.6), the multiplier estimate on ${\mathcal {J}}_k$ (A.13), the paraproduct estimate (A.8), and the $\flat$-estimates (4.19),

Hence, by Cauchy–Schwarz with respect to the finite measure $\frac{dk}{\langle k \rangle ^{1+\kappa }}$, the potential bound (5.9), and Young’s inequality,

(5.49)

For the remaining term in (5.48), note that by the trivial embedding $H^1 \hookrightarrow H^{1-\kappa }$ and the bound (4.18) applied to $R_{k'}$,

(5.50)

Inserting (5.49) and (5.50) into (5.48) establishes (5.22).

5.6.2 Proof of (5.45)

We start with the first integral in ${\mathcal {R}}^{a,1}_K$. Fix $\kappa > 0$ and let q be such that $(1+\kappa )^{-1} + q^{-1} = 1$. Then, by Young’s inequality (and remembering $\beta > 1$),

(5.51)

Adding and subtracting $g_K$ into the second term on the righthand side and using the pointwise potential bound (5.8),

(5.52)

where we recall that $|\cdot |_\infty$ is the supremum norm.

By the bounds on $g_K$ (5.11) and $V_K$ (5.23), taking $\kappa <1$ yields

(5.53)

Above, we recall that $N^\Xi _K$ can contain constants $C=C(\eta )>0$.

For the remaining term on the righthand side of (5.52), we reorganise terms and iterate the preceding argument:

(5.54)

provided that $\kappa < \frac{1}{4}$.

We now estimate the second integral in ${\mathcal {R}}^{a,1}_K$. Let ${\tilde{\kappa }} > 0$ be sufficiently small. Let q be such that $\frac{3-{\tilde{\kappa }}}{4(1+{\tilde{\kappa }})(1-{\tilde{\kappa }})} + \frac{1}{q} = 1$. Moreover, let $\theta = \frac{2{\tilde{\kappa }}(1-{\tilde{\kappa }})}{(1+{\tilde{\kappa }}) (1-2{\tilde{\kappa }})}$. By duality (A.1), the fractional Leibniz rule (A.2) and interpolation (A.4),

(5.55)

By the change of variables (5.21) in reverse, reorganising terms, Young’s inequality, the bound on $U_K$ (5.22), and using $\varepsilon < 1$,

(5.56)

For the last term on the righthand side of (5.56), we iterate the potential bound (5.8) and bound on $g_K$ (5.11) as in the estimate of (5.52):

(5.57)

where in the penultimate line we used Young’s inequality and in the last line we have used (5.23).

Combining (5.51), (5.53), (5.54), (5.56), and (5.57) establishes (5.45).

5.6.3 Proof of (5.46)

For any $\theta \in (0,1)$ let $\frac{1}{p} = \frac{\theta }{4} + \frac{1-\theta }{2}$ and let $\frac{1}{p'} = 1- \frac{1}{p}$. Then, by duality (A.1), the paraproduct estimate (A.8), the Bernstein-type bounds on the derivatives of the drift (4.20), and bounds on the $\partial _k g_k^\flat$ (5.14),

(5.58)

where in the last inequality we have reordered terms.

Then,

(5.59)

where in the first line we have used Bernstein’s inequality (A.5); in the second line we have used interpolation (A.4); in the penultimate line we used Young’s inequality; and in the last line we have used the bounds on $V_K$ (5.23), $U_k$ (5.22), together with the potential bound (5.9).

In order to bound the second integrand in ${\mathcal {R}}^{a,2}_K$, we use Fubini’s theorem, the Cauchy–Schwarz inequality, the bounds on $V_k^\flat$ (4.19) and $\partial _k V_K^\flat$ (4.20), and the bounds on $g_K$ (5.11) to obtain

(5.60)

where in the last inequality we have used the observation made in Remark 4.3 that $|\gamma _k|\lesssim \log \langle k \rangle$.

Thus, by Young’s inequality (applied to each term after expanding the sum), the potential bound (5.9), and the bound on $V_K$ (5.23),

(5.61)

Combining (5.59) and (5.61) yields (5.46).

5.6.4 Proof of (5.47)

We write ${\mathcal {R}}^{a,3}_K = I_1 + I_2 + I_3$, where

Let $\theta \in (0,1)$ be sufficiently small and let $\frac{1}{p} = \frac{\theta }{4} + \frac{1-\theta }{2}$, $\frac{1}{q} = \frac{1-\theta }{2}$ and $\frac{1}{p'} = \frac{1}{2}-\frac{1}{p}$, $\frac{1}{q'} = \frac{1}{2} - \frac{1}{q}$. Then,

(5.62)

where the first inequality is by duality (A.1); the second inequality is by the commutator estimate (A.14) and the triangle inequality; and the third inequality is by the multiplier estimate (A.13) and the paraproduct estimate (A.8).

Thus,

(5.63)

where the first inequality is by Bernstein’s inequality (A.5); and the second inequality is by the $\flat$-bounds applied to $V_k^\flat + g_k^\flat$ (4.19), interpolation (A.4), and the trivial bound $\Vert V_K + g_K \Vert _{B^{4\kappa \theta }_{4,\infty }} \lesssim \Vert V_K+g_K\Vert _{L^4}$.

By applying Young’s inequality, the potential bound (5.9), and the bound on $V_K$ (5.23), we have

(5.64)

Now consider $I_2$. Using the commutator estimate (A.10) with , $g = V_k^\flat + g_k^\flat$ and , followed by the paraproduct estimate (A.8), we obtain

(5.65)

By applying Young’s inequality, the potential bound (5.9), and the a priori bound on $V_K$ (5.23),

(5.66)

where the final inequality uses the multiplier estimate (A.13), the $\flat$-bounds applied to $V_K+g_K$ (4.19), and Bernstein’s inequality (A.6).

For $I_3$, we apply duality (A.1), the commutator estimate (A.11) with and $g=V_k^\flat + g_k^\flat$, followed by the $\flat$-bounds applied to $V_K+g_K$ (4.19), to obtain

(5.67)

where in the last line we have used Young’s inequality, the potential bound (5.9), and the bound on $V_K$ (5.23) as in (5.66).

Using that ${\mathcal {R}}^{a,3}_K = I_1 + I_2 + I_3$, the estimates (5.64), (5.66), and (5.67) establish (5.47).

5.7 A lower bound on the effective Hamiltonian

The following lemma, based on [GJS76b, Theorem 3.1.1], gives a $\beta$-independent lower bound on ${\mathcal {H}}^{\mathrm{eff}}_K(Z_K)$ in terms of the $L^2$-norm of the fluctuation field $Z_K^\perp = Z_K - Z_K$, where we recall and for . This is useful for us because the latter can be bounded in a $\beta$-independent way (see Sect. 5.8.1).

Lemma 5.21

There exists $C>0$ such that, for any $\zeta > 0$ and $K \in (0,\infty )$,

$$\begin{aligned} {\mathcal {H}}^{\mathrm{eff}}_K (Z_K) \geqslant -CN^3 -\zeta \int _{{\mathbb {T}}_N}\big (Z_K^\perp \big )^2 dx \end{aligned}$$

(5.68)

provided $\eta < \min \Big ( \frac{1}{32}, \frac{2\zeta }{49} \Big )$.

Proof

First, we write

Fix . Without loss of generality, assume $\sigma (x) = 1$ and, hence, $h(x) = {\sqrt{\beta }}$. Define

$$\begin{aligned} I(x) = \frac{1}{2} {\mathcal {V}}_\beta (Z_K(x)) - \frac{\eta }{2} (Z_K(x)-{\sqrt{\beta }})^2 - \log \chi _{+}(Z_K(x)). \end{aligned}$$

In order to show (5.68), it suffices to show that, for some $C > 0$,

$$\begin{aligned} I(x) + \zeta Z_K^\perp (x)^2 \geqslant -C. \end{aligned}$$

The fundamental observation is that $Z_K(x) \mapsto \frac{1}{2}{\mathcal {V}}_\beta (Z_K(x))$ can be approximated from below near the minimum at $Z_K(x) = {\sqrt{\beta }}$ by the quadratic $Z_K(x) \mapsto \frac{\eta }{2} (Z_K(x)-{\sqrt{\beta }})^2$ provided $\eta$ is taken sufficiently small. Indeed, we have

$$\begin{aligned} \frac{1}{2}{\mathcal {V}}_\beta (Z_K(x)) - \frac{\eta }{2} (Z_K(x) - {\sqrt{\beta }})^2&= \frac{1}{2\beta }(Z_K(x) -{\sqrt{\beta }})^2 \Big ( (Z_K(x) + {\sqrt{\beta }})^2 - \eta \beta \Big ) \end{aligned}$$

which is non-negative provided $|Z_K(x) + {\sqrt{\beta }}| \geqslant \sqrt{\eta \beta }$. Thus, this approximation is valid except for the region near the opposite potential well satisfying $(-1-\sqrt{\eta }){\sqrt{\beta }}< Z_K(x) < (-1+\sqrt{\eta }){\sqrt{\beta }}$ (see Figure 2). When $Z_K(x)$ sits in this region, we split $Z_K(x) = Z_K(x) + Z_K^\perp (x)$ and observe that:

• either the deviation to the opposite well is caused by $Z_K(x)$, which is penalised by the logarithm in I(x);

• or, the deviation is caused by $Z_K^\perp (x)$, which produces the integral involving $Z_K^\perp$ in (5.68).

Fig. 2

Plot of ${\mathcal {V}}_\beta (Z_K(x))$ and $\frac{\eta }{2} (Z_K(x) - {\sqrt{\beta }})^2$

Motivated by these observations, we split the analysis of I(x) into two cases. First we treat the case $Z_K(x) \in {\mathbb {R}}{\setminus }\Big ( -\frac{4{\sqrt{\beta }}}{3}, -\frac{2{\sqrt{\beta }}}{3} \Big )$. Under this condition, we have

$$\begin{aligned} \frac{1}{2} {\mathcal {V}}_\beta (Z_K(x)) \geqslant \eta (Z_K(x)-{\sqrt{\beta }})^2 \end{aligned}$$

provided that $\eta \leqslant \frac{1}{9}$. Since $\chi _+(\cdot ) \leqslant 1$, $-\log \chi _+(\cdot ) \geqslant 0$. It follows that $I(x) \geqslant 0$.

Now let $Z_K(x) \in \Big ( -\frac{4{\sqrt{\beta }}}{3}, -\frac{2{\sqrt{\beta }}}{3} \Big )$. Necessarily, either $Z_K(x) \leqslant - \frac{{\sqrt{\beta }}}{3}$ or $Z_K^\perp (x) \leqslant - \frac{{\sqrt{\beta }}}{3}$.

We first assume that $Z_K(x) \leqslant - \frac{{\sqrt{\beta }}}{3}$. By standard bounds on the Gaussian error function (see e.g. [GJS76b, Lemma 2.6.1]), for any $\theta \in (0,1)$ there exists $C=C(\theta )>0$ such that

Applying this with $\theta \in (\frac{1}{2}, 1)$ and that, by our assumption, $Z_K(x) - {\sqrt{\beta }}> 4Z_K(x)$,

provided $\eta < \min \Big ( \zeta , \frac{1}{32} \Big )$.

Finally, assume that $Z_K^\perp (x) < - \frac{{\sqrt{\beta }}}{3}$. Since $Z_K(x) - {\sqrt{\beta }}\in \Big ( -\frac{7{\sqrt{\beta }}}{3}, -\frac{-5{\sqrt{\beta }}}{3} \Big )$, we have

$$\begin{aligned} \begin{aligned} I(x) + \zeta (Z_K^\perp (x))^2&\geqslant - \frac{49\eta }{18} \beta + \zeta (Z_K^\perp (x))^2 \geqslant 0 \end{aligned} \end{aligned}$$

(5.69)

provided that $\eta \leqslant \frac{2\zeta }{49}$. $\quad \square$

5.8 Proof of Proposition 5.3

5.8.1 Proof of the lower bound on the free energy (5.1)

We derive bounds uniform in $\sigma$ for each term in the expansion (5.10). Since there are $2^{N^3}$ terms, this is sufficient to establish (5.1). Fix $\sigma \in \{\pm 1\}^{{\mathbb {B}}_N}$.

Recall

$$\begin{aligned} -\log {\mathscr {Z}}_{\beta ,N,K}^\sigma = -\log {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^\sigma } + F^\sigma _{\beta ,N,K}. \end{aligned}$$

(5.70)

Let $C_P > 0$ be the sharpest constant in the Poincaré inequality (A.15) on unit boxes. Note that $C_P$ is independent of N. Fix $\zeta < \frac{1}{8C_P}$ and let $\varepsilon = 1-8C_P\zeta > 0$. By Proposition 5.11 and Lemma 5.21 there exists $C=C(\zeta ,\eta )>0$ such that, for every $v \in {\mathbb {H}}_{b,K}$,

provided $\eta< \frac{2\zeta }{49} < \frac{1}{196 C_P}$.

Note that for any $f \in L^2$, $\int _{{\mathbb {T}}_N}(f^\perp )^2 dx \leqslant \int _{{\mathbb {T}}_N}f^2 dx$. Therefore, using the inequality $(a_1 + a_2 + a_3 + a_4)^2 \leqslant 4(a_1^2 + a_2^2 + a_3^2 + a_4^2)$ and that $Z_K^\perp (x) = (V_K+g_K)^\perp (x)$, we have

Arguing as in (5.49),

By the Poincaré inequality (A.15) on unit boxes,

where in the last inequality we used that $\int _{{\mathbb {T}}_N}|\nabla R_K|^2 dx \leqslant \Vert R_K\Vert _{H^1}^2$ and Lemma 4.11 (applied to $R_K$).

Similarly, by the Poincaré inequality (A.15) and the (trivial) bound $\Vert \nabla g_K\Vert _{L^2}^2 \leqslant \Vert \nabla {\tilde{g}}_K \Vert _{L^2}^2$ (5.12),

$$\begin{aligned} \int _{{\mathbb {T}}_N}(g_K^\perp )^2 dx \leqslant C_P\int _{{\mathbb {T}}_N}|\nabla g_K|^2 dx \leqslant C_P\int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx. \end{aligned}$$

Then, recalling that $\beta > 1$,

$$\begin{aligned} {\mathbb {E}}\Psi _K(v)&\geqslant {\mathbb {E}}\Bigg [ - C N^\Xi _K + 4\zeta C_P\Big ( 1 - \frac{1}{\beta ^3} \Big ) \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx \\&\quad \quad \quad + \Big ( 4\zeta C_P - 4\zeta C_P \Big ) \int _{{\mathbb {T}}_N}\int _0^Kr_k^2 dk dx - 4\zeta C_P \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx \Bigg ] \\&\geqslant {\mathbb {E}}\Bigg [ - C N^\Xi _K - 4\zeta C_P \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx \Bigg ] \end{aligned}$$

from which, by Proposition 4.7, we obtain

$$\begin{aligned} -\log {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^\sigma } \geqslant -CN^3 -4\zeta C_P \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx. \end{aligned}$$

Inserting this into (5.70) and using that $F_{\beta ,N,K}^\sigma \geqslant \frac{1}{2} \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx$ (see (5.15)) yields:

$$\begin{aligned} -\log {\mathscr {Z}}_{\beta ,N,K}^\sigma \geqslant -CN^3 + \Big ( \frac{1}{2} - 4\zeta C_P \Big ) \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx \geqslant -CN^3 \end{aligned}$$

which establishes (5.1).

5.8.2 Proof of the upper bound on the free energy (5.2)

We (globally) translate the field to one of the minima of ${\mathcal {V}}_\beta$: this kills the constant $\beta$ term. Thus, under the translation $\phi = \psi + {\sqrt{\beta }}$,

$$\begin{aligned} {\mathscr {Z}}_{\beta ,N,K} = {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^+(\psi _K)} \end{aligned}$$

where

$$\begin{aligned} {\mathcal {H}}_{\beta ,N,K}^+(\psi _K)&= \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta ^+(\psi _K) - \frac{\gamma _K}{\beta ^2}:(\psi _K + \sqrt{\beta })^2: - \delta _K - \frac{\eta }{2} :\psi _K^2:dx \end{aligned}$$

and

$$\begin{aligned} {\mathcal {V}}_\beta ^+(a) = \frac{1}{\beta } a^2(a+2{\sqrt{\beta }})^2 =\frac{1}{\beta }a^4 + \frac{4}{{\sqrt{\beta }}}a^3 + 4a^2. \end{aligned}$$

We apply the Proposition 4.7 to ${\mathscr {Z}}_{\beta ,N,K}$ with the infimum taken over ${\mathbb {H}}_K$. In order to obtain an upper bound, we choose a particular drift in the corresponding stochastic control problem (4.13). Following [BG19], we seek a drift that satisfies sufficient moment/integrability conditions with estimates that are extensive in $N^3$, as formalised in Lemma 5.22 below. Such a drift is constructed using a fixed point argument, hence the need to work in the Banach space ${\mathbb {H}}_K$ as opposed to ${\mathbb {H}}_{b,K}$.

Lemma 5.22

There exist processes and satisfying and a unique fixed point ${\check{v}} \in {\mathbb {H}}_K$ of the equation

(5.71)

where ${\check{V}}_K = \int _0^K{\mathcal {J}}_k {\check{v}}_k dk$, such that the following estimate holds: for all $p \in [1,\infty )$, there exists $C=C(p,\eta )>0$ such that, for all $\beta > 1$,

$$\begin{aligned} {\mathbb {E}}\Bigg [ \int _{{\mathbb {T}}_N}|{\check{V}}_K|^p dx + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^K{\check{r}}_k^2 dk dx \Bigg ] \leqslant CN^3 \end{aligned}$$

(5.72)

where .

Proof

See [BG19, Lemma 6]. Note that the key difficulty lies in obtaining the right N dependence in (5.72). Due to the paraproduct in the definition of (5.71), one can show that this requires finding a decomposition of such that has Besov-Hölder norm that is uniformly bounded in $N^3$ (see Proposition A.5). Such a bound is not true for (see Remark 4.5). This is overcome by defining to be a random truncation of the Fourier series of , where the location of the truncation is chosen to depend on the Besov-Hölder norm of . $\quad \square$

For $v \in {\mathbb {H}}_K$, let

and define ${\mathcal {R}}_K^+$ by

$$\begin{aligned} \Psi _K^+(v) = {\mathcal {R}}_K^+ - \frac{\eta }{2} \int _{{\mathbb {T}}_N}V_K^2 dx + \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta ^+(V_K) dx + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^Kv_k^2\ dk dx. \end{aligned}$$

We observe

$$\begin{aligned} \Psi _K^+(v) \leqslant {\mathcal {R}}_K^+ + \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta ^+(V_K) dx + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^Kv_k^2 dkdx. \end{aligned}$$

(5.73)

Thus, unlike the lower bound, the negative mass $-\frac{\eta }{2} \int _{{\mathbb {T}}_N}V_K^2 dx$ can be ignored in bounding the upper bound on the free energy.

Now fix ${\check{v}}$ as in (5.71). Arguing as in Proposition 5.11, there exists ${\tilde{{\mathcal {R}}}}^+_K$ such that

$$\begin{aligned} {\mathcal {R}}^+_K + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^K{\check{v}}_k^2 dk dx \approx {\tilde{{\mathcal {R}}}}_K^+ + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^K{\check{r}}_k^2 dk dx \end{aligned}$$

(5.74)

and ${\tilde{{\mathcal {R}}}}_K^+$ satisfies the following estimate: for every $\varepsilon > 0$, there exists $C=C(\varepsilon ,\eta ) > 0$ such that, for all $\beta > 1$,

$$\begin{aligned} |{\tilde{{\mathcal {R}}}}_K^+| \leqslant C N^\Xi _K + \varepsilon \Big ( \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta ^+({\check{V}}_K) dx + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^K{\check{r}}_k^2 dk dx \Big ). \end{aligned}$$

(5.75)

Above, we have used that the moment conditions (5.72) are sufficient for conclusions of Lemma 5.14 to apply to ${\check{v}}$.

Thus, by (5.73), (5.74), and (5.75),

$$\begin{aligned} {\mathbb {E}}[\Psi _K^+({\check{v}})] \leqslant CN^3 + (1+\varepsilon ) {\mathbb {E}}\Big [ \int _{{\mathbb {T}}_N}{\mathcal {V}}_{\beta }^+({\check{V}}_K) + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^K{\check{r}}_k^2 dk dx \Big ]. \end{aligned}$$

(5.76)

By Young’s inequality, $\frac{1}{\beta }a^4 + \frac{4}{{\sqrt{\beta }}}a^3 + 4a^2 \leqslant 3a^4 + 6a^2 \leqslant 9a^4 + 9$ for all $\beta > 1$ and $a \in {\mathbb {R}}$. Thus,

$$\begin{aligned} \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta ^+({\check{V}}_K) dx \leqslant 9 \int _{{\mathbb {T}}_N}{\check{V}}_K^4 dx + 9N^3. \end{aligned}$$

Inserting this into (5.76) and using the moment estimates on the drift (5.72) yields

$$\begin{aligned} {\mathbb {E}}[\Psi _K^+({\check{v}})] \leqslant CN^3 + (1+\varepsilon ) {\mathbb {E}}\Big [ 9 \int _{{\mathbb {T}}_N}{\check{V}}_K^4 dx + \frac{1}{2} \int _{{\mathbb {T}}_N}\int _0^K{\check{r}}_k^2 dk dx \Big ] \leqslant CN^3. \end{aligned}$$

Hence, by Proposition 4.7,

$$\begin{aligned} -\log {\mathscr {Z}}_{\beta ,N,K} = \inf _{v \in {\mathbb {H}}_K} {\mathbb {E}}\Psi ^+_K(v) \leqslant {\mathbb {E}}\Psi ^+_K({\check{v}}) \leqslant CN^3 \end{aligned}$$

thereby establishing (5.2).

5.9 Proof of Proposition 5.1

We begin with two propositions, the first of which is a type of Itô isometry for fields under ${\nu _{\beta ,N}}$ and the second of characterises functions against which the Wick square field can be tested against. Together, they imply that the random variables in Proposition 5.1 are integrable and that these expectations can be approximated using the cutoff measures $\nu _{\beta ,N,K}$. Recall also Remarks 3.1 and 3.3 .

Proposition 5.23

Let $f \in H^{-1+\delta }$ for some $\delta > 0$. For every $K \in (0,\infty )$, let $\phi ^{(K)} \sim \nu _{\beta ,N,K}$ and $\phi \sim {\nu _{\beta ,N}}$.

The random variables $\{ \int _{{\mathbb {T}}_N}\phi ^{(K)} f dx \}_{K > 0}$ converge weakly as $K \rightarrow \infty$ to a random variable

$$\begin{aligned} \phi (f) = \int _{{\mathbb {T}}_N}\phi f dx \in L^2({\nu _{\beta ,N}}). \end{aligned}$$

Moreover, for every $c> 0$,

$$\begin{aligned} \big \langle \exp {\big (c\phi (f)^2\big )} \big \rangle _{\beta ,N} < \infty . \end{aligned}$$

Proof

Let $\{ f_n \}_{n \in {\mathbb {N}}} \subset C^\infty ({\mathbb {T}}_N)$ such that $f_n \rightarrow f$ in $H^{-1+\delta }$. We first show that $\{ \phi (f_n) \}$ is Cauchy in $L^2({\nu _{\beta ,N}})$.

Let $\varepsilon > 0$. Choose $n_0$ such that, for all $n,m > n_0$, $\Vert f_n - f_m \Vert _{H^{-1+\delta }} < \frac{\varepsilon }{N^3}$.

Fix $n,m > n_0$ and let $\delta f = f_n - f_m$. Then,

$$\begin{aligned} |\phi (f_n) - \phi (f_m)|^2 = \varepsilon \cdot \frac{1}{\varepsilon }\phi (\delta f)^2 \leqslant \varepsilon e^{\frac{1}{\varepsilon }\phi (\delta f)^2 }. \end{aligned}$$

(5.77)

By Proposition 5.3, there exists $C=C(\eta )>0$ such that

$$\begin{aligned} \Big \langle e^{\frac{1}{\varepsilon }\phi (\delta f)^2 } \Big \rangle _{\beta ,N}&= \lim _{K \rightarrow \infty } \frac{1}{{\mathscr {Z}}_{\beta ,N,K}} {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K) + \frac{1}{\varepsilon }\phi _K(\delta f)^2} \\&\leqslant e^{CN^3} \limsup _{K \rightarrow \infty }{\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K) + \frac{1}{\varepsilon }\phi _K(\delta f)^2}. \end{aligned}$$

We apply Proposition 4.7 to the expectation on the righthand side (with total energy cutoff suppressed, see Remark 4.8 and the paragraph that precedes it).

For $v \in {\mathbb {H}}_{b,K}$, define

Expanding out the second term (and ignoring the prefactor $\frac{1}{\varepsilon }$ for the moment), we obtain:

(5.78)

Consider the first integral in (5.78). By Parseval’s theorem, the Fourier coefficients of (see (4.2)), and Itô’s isometry,

(5.79)

where sums are taken over frequencies $n_i \in (N^{-1}{\mathbb {Z}})^3$. Above, the N dependency in the last inequality is due to our Sobolev spaces being defined with respect to normalised Lebesgue measure .

For the second term in (5.78), by Parseval’s theorem, Itô’s isometry, and the Fourier coefficients of (see (4.6)), we obtain

(5.80)

For the final term in (5.78), by duality (A.1)

$$\begin{aligned} \Big (\int _{{\mathbb {T}}_N}U_K \delta f dx \Big )^2 \leqslant N^6 \Vert \delta f \Vert _{H^{-1+\delta }}^2 \Vert U_K \Vert _{H^{1-\delta }}^2. \end{aligned}$$

(5.81)

Therefore, using that $\Vert \delta f \Vert _{H^{1-\delta }}^2 \leqslant \frac{\varepsilon ^2}{N^6}$, the estimates (5.79), (5.80), and (5.81) yield:

(5.82)

Using arguments in Sect. 5.8.1, it is straightforward to show that there exists $C=C(\eta ,\beta )>0$ such that, for $\varepsilon$ sufficiently small,

$$\begin{aligned} {\mathbb {E}}\Psi _K^{\delta f}(v) \geqslant - CN^3 \end{aligned}$$

for every $v \in {\mathbb {H}}_{b,K}$ (note that $\beta$ dependence is not important here).

Inserting this into Proposition 4.7 gives

$$\begin{aligned} \limsup _{K \rightarrow \infty } \langle e^{-{\mathcal {H}}_{\beta ,N,K}(\phi _K) + \frac{1}{\varepsilon } \phi _K(\delta f)^2} \rangle _{\beta ,N,K} \leqslant e^{CN^3}. \end{aligned}$$

(5.83)

Taking expectations in (5.77) and using (5.83) finishes the proof that $\{ \phi (f_n) \}$ is Cauchy in $L^2({\nu _{\beta ,N}})$.

Similar arguments can be used to show exponential integrability of the limiting random variable, $\phi (f)$ and that,

$$\begin{aligned} \sup _{K > 0} |\langle |\phi ^{(K)}(f_n) - \phi ^{(K)}(f)| \rangle _{\beta ,N,K} \rightarrow 0 \quad \text {as } n \rightarrow \infty . \end{aligned}$$

We now show that $\phi ^{(K)}(f)$ converges weakly to $\phi (f)$ as $K \rightarrow \infty$. Let $G : {\mathbb {R}}\rightarrow {\mathbb {R}}$ be bounded and Lipschitz with Lipschitz constant $|G|_{\mathrm{Lip}}$, and let $\varepsilon > 0$. Choose n sufficiently large so that

$$\begin{aligned} \sup _{K > 0} |\langle |\phi ^{(K)}(f_n) - \phi ^{(K)}(f)| \rangle _{\beta ,N,K} < \frac{\varepsilon }{2|G|_{\mathrm{Lip}}} \end{aligned}$$

and

$$\begin{aligned} \langle |\phi (f_n)) - \phi (f)| \rangle _{\beta ,N} < \frac{\varepsilon }{2|G|_{\mathrm{Lip}}}. \end{aligned}$$

Then,

$$\begin{aligned} |\langle G(\phi ^{(K)}(f)) \rangle _{\beta ,N,K} - \langle G(\phi (f)) \rangle _{\beta ,N}&\leqslant \sup _{K > 0} |\langle G(\phi ^{(K)}(f_n)) - G(\phi ^{(K)}(f)) \rangle _{\beta ,N,K} | \\&\quad + | \langle G(\phi ^{(K)}(f_n)) \rangle _{\beta ,N,K} - \langle G(\phi (f_n)) \rangle _{\beta ,N}| \\&\quad + |\langle G(\phi (f_n)) - G(\phi (f)) \rangle _{\beta ,N}| \\&\leqslant | \langle G(\phi ^{(K)}(f_n)) \rangle _{\beta ,N,K} - \langle G(\phi (f_n)) \rangle _{\beta ,N}| + \varepsilon . \end{aligned}$$

The first term on the righthand side goes to zero as $K \rightarrow \infty$ since $f_n \in C^\infty$. Thus,

$$\begin{aligned} \lim _{K \rightarrow \infty } |\langle G(\phi (f)) \rangle _{\beta ,N,K} - \langle G(\phi (f)) \rangle _{\beta ,N} \leqslant \varepsilon . \end{aligned}$$

Since $\varepsilon$ is arbitrary, we have shown that $\phi ^{(K)}(f)$ converges weakly to $\phi (f)$. $\quad \square$

Proposition 5.24

Let $f \in B^s_{\frac{4}{3},1} \cap L^{2}$ for some $s > \frac{1}{2}$. For every $K \in (0,\infty )$, let $\phi ^{(K)} \sim \nu _{\beta ,N,K}$ and $\phi \sim {\nu _{\beta ,N}}$.

The random variables $\{ \int _{{\mathbb {T}}_N}:(\phi ^{(K)})^2: f dx \}_{K>0}$ converge weakly as $K \rightarrow \infty$ to a random variable

$$\begin{aligned} :\phi ^2:(f) = \int _{{\mathbb {T}}_N}:\phi ^2: f dx \in L^2({\nu _{\beta ,N}}). \end{aligned}$$

Moreover, for $c > 0$,

$$\begin{aligned} \big \langle \exp \big ( c:\phi ^2:(f) \big ) \big \rangle _{\beta ,N} < \infty . \end{aligned}$$

Proof

The proof of Proposition 5.24 follows the same strategy as the proof of Proposition 5.23, so we do not give all the details. The only real key difference is the analytic bounds required in the stochastic control problem. Indeed, these require one to tune the integrability assumptions on f in order to get the required estimates.

It is not too difficult to see that the term we need to control is the integral

(5.84)

Strictly speaking, we need to control the above integral with f replaced by $\delta f = f_n - f_m$, where $\{ f_n \}_{n \in {\mathbb {N}}}\subset C^\infty ({\mathbb {T}}_N)$ such that $f_n \rightarrow f$ in $B^s_{\frac{3}{4}, 1} \cap L^2$, but the analytic bounds are the same.

Note that . Moreover by Young’s inequality and the additional integrability assumption $f \in L^2$, for any $\varepsilon > 0$ we have

$$\begin{aligned} \int _{{\mathbb {T}}_N}V_K^2 f dx \lesssim \frac{1}{\varepsilon } \int _{{\mathbb {T}}_N}f^2 dx + \varepsilon \int _{{\mathbb {T}}_N}V_K^4 dx \end{aligned}$$

which can be estimated as in the proof of Proposition 5.23. Thus, we only need to estimate the second integral in (5.84). Note that the product is a well-defined distribution from a regularity perspective as $K \rightarrow \infty$ since $f \in B^{s}_{\frac{4}{3},1}$ for $s > \frac{1}{2}$. The difficulty in obtaining the required estimates comes from integrability issues.

We split the integral into three terms by using the paraproduct decomposition . The integral associated to is straightforward to estimate, so we focus on the first two terms. Since $f \in L^2$ and , by the paraproduct estimate A.8 we have . Thus, the integral can be treated similarly as in the proof of Proposition 5.23. Note that, in this proposition the use of Hölder-Besov norms is fine because we are not concerned with issues of N dependency. Moreover, note that if we just used that $f \in B^s_{\frac{4}{3},1}$ the resulting integrability of is not sufficient to justify testing against $V_K$, which can be bounded in $L^2$-based Sobolev spaces. For the final integral, by the resonant product estimate (A.9) we have . Hence, we can use Young’s inequality to estimate and then argue similarly as in the proof of Proposition 5.23. $\quad \square$

Without loss of generality, we assume $a_0 = a = 1$ in Proposition 5.1 and we split its proof into Lemmas 5.25, 5.26, and 5.27 .

Lemma 5.25

There exists $\beta _0 > 1$ and $C_Q > 0$ such that, for any $\beta > \beta _0$,

Proof

For any $K \in (0,\infty )$, define

$$\begin{aligned} {\mathcal {H}}_{\beta ,N,K}^{Q_1}(\phi _K) = \int _{{\mathbb {T}}_N}:{\mathcal {V}}_\beta ^{Q_1}(\phi _K): - \frac{\gamma _K}{\beta ^2}:\phi _K^2: - \delta _K - \frac{\eta }{2} :\phi _K^2: dx \end{aligned}$$

where

$$\begin{aligned} {\mathcal {V}}^{Q_1}_\beta (a) = {\mathcal {V}}_\beta (a) - \frac{1}{{\sqrt{\beta }}}(\beta - a^2) - \frac{1}{4} = \frac{1}{\beta }\Bigg (a^2 - \Big (\beta + \frac{{\sqrt{\beta }}}{2} \Big ) \Bigg )^2. \end{aligned}$$

Then, by Propositions 5.24 and 5.3 , there exists $C=C(\eta )>0$ such that

where

Therefore, we have reduced the problem to proving Proposition 5.3 for the potential ${\mathcal {V}}_\beta ^{Q_1}$ instead of ${\mathcal {V}}_\beta$. The proof follows essentially word for word after two observations: first, the same $\gamma _K$ and $\delta _K$ works for both ${\mathcal {V}}_\beta$ and ${\mathcal {V}}_\beta ^{Q_1}$ since the quartic term is unchanged. Second, since $\sqrt{\beta + \frac{{\sqrt{\beta }}}{2}} = {\sqrt{\beta }}+ o({\sqrt{\beta }})$ as $\beta \rightarrow \infty$, the treatment of $\beta$-dependence of the estimates in Sect. 5.6 is exactly the same. $\quad \square$

Lemma 5.26

There exists $\beta _0 > 1$ and $C_Q > 0$ such that, for any $\beta > \beta _0$,

(5.85)

Proof

By Propositions 5.23, 5.24, and 5.3 , there exists $C=C(\eta )>0$ such that, for $\beta$ sufficiently large,

where

As in Sect. 5.2, we perform the expansion

$$\begin{aligned} -\log {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^{Q_2}(\phi _K)} = \sum _{\sigma \in \{ \pm 1 \}^{{\mathbb {B}}_N}} e^{-F_{\beta ,N,K}^\sigma }{\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^{Q_2,\sigma }(\phi _K)} \end{aligned}$$

(5.86)

where $F_{\beta ,N,K}^\sigma$ is defined in (5.15) and

Fix $\sigma \in \{\pm 1\}^{{\mathbb {B}}_N}$. For $v \in {\mathbb {H}}_{b,K}$, define

(5.87)

where $\Psi _K=\Psi _K^\sigma$ is defined in (5.17).

We estimate second term in (5.87). First, note that

By a standard Gaussian covariance calculation, there exists $C=C(\eta )>0$ such that

For the other term, by the Cauchy–Schwarz inequality followed by bounds on the potential (5.8) and $g_K$ (5.11), the following estimate holds: for any $\zeta > 0$,

where $C_P > 0$ is the Poincaré constant on unit boxes (A.15).

We now estimate the third term in (5.87). Since ,

(5.88)

For the first integral on the righthand side of (5.88), by change of variables (5.21), and the paraproduct decomposition (A.7), we have

Note that and can be respectively bounded uniformly in ${\mathcal {C}}^{-\frac{1}{2} - \kappa }$ and ${\mathcal {C}}^{-2\kappa }$ by using the paraproduct estimate (A.8) and a mild modification for the second term. Moreover, by Proposition 4.4, can be bounded uniformly in ${\mathcal {C}}^{-2\kappa }$. Hence, by (5.88), Proposition 4.4, duality (A.1), the potential bounds (5.8), and the bounds on $U_K$ (5.22), for any $\varepsilon > 0$ there exists $C=C(\varepsilon ,\eta )>0$ such that

For the second integral on the righthand side of (5.88), again by (5.8) and (5.11), there exists an inessential constant $C>0$ such that

$$\begin{aligned} \int _{{\mathbb {T}}_N}\frac{1}{{\sqrt{\beta }}}(V_K+g_K)^2 dx \leqslant CN^3 + \zeta C_P \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx. \end{aligned}$$

Arguing as in Sect. 5.8.1 and taking into account the calculations above, the following estimate holds: let $\zeta < \frac{1}{8C_P}$ and $\varepsilon = 1-8C_P\zeta > 0$ as in Sect. 5.8.1. Then, provided $\eta < \frac{1}{196C_P}$ and $\beta > 1$,

$$\begin{aligned} {\mathbb {E}}\Psi _K^{Q_2}(v)&\geqslant {\mathbb {E}}\Bigg [ -C(\varepsilon ,\zeta ,\eta ) N^\Xi _K + \Big (\frac{1-\varepsilon }{2} - \frac{4C_P\zeta }{2\beta ^3} - 2C_P \zeta \Big ) \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx \\&\quad \quad \quad + \Big (\frac{1-\varepsilon }{2} - 4\zeta C_P \Big ) \int _{{\mathbb {T}}_N}\int _0^Kr_k^2 dk dx - 4\zeta C_P \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K |^2 dx \Bigg ] \\&\geqslant -CN^3 -4\zeta C_P \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx. \end{aligned}$$

Hence, by Proposition 4.7 applied with the Hamiltonian ${\mathcal {H}}_{\beta ,N,K}^{Q_2,\sigma }(\phi _K)$ with total energy cutoff suppressed (see Remark 4.8),

$$\begin{aligned} F_{\beta ,N,K}^\sigma - \log {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^{Q_2,\sigma }} \geqslant -CN^3 + \Big ( \frac{1}{2} - 4\zeta C_P\Big ) \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx \geqslant - CN^3 \end{aligned}$$

This estimate is uniform in $\sigma$, thus summing over the $2^{N^3}$ terms in the expansion (5.86) yields (5.85). $\quad \square$

Lemma 5.27

There exists $\beta _0 > 1$ and $C_Q > 0$ such that, for any $\beta > \beta _0$,

(5.89)

where B is a set of unordered pairs of nearest-neighbour blocks that partitions ${{\mathbb {B}}_N}$.

Proof

By Propositions 5.23 and 5.3 there exists $C=C(\eta )>0$ such that, for $\beta$ sufficiently large,

where

We expand

$$\begin{aligned} -\log {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^{Q_3}(\phi _K)} = \sum _{\sigma \in \{ \pm 1 \}^{{\mathbb {B}}_N}} e^{-F_{\beta ,N,K}^\sigma }{\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^{Q_3,\sigma }} \end{aligned}$$

where $F_{\beta ,N,K}^\sigma$ is defined in (5.15) and

Fix $\sigma \in \{\pm 1\}^{{\mathbb {B}}_N}$. For $v \in {\mathbb {H}}_{b,K}$, define

where $\Psi _K(v) = \Psi _K^\sigma (v)$ is defined in (5.17).

A standard Gaussian calculation yields for some constant $C=C(\eta )>0$. Hence, by the triangle inequality, Proposition 4.4 and the Cauchy–Schwarz inequality,

The integral with the paraproduct can be estimated as in (5.49) to establish: for any $\zeta > 0$,

where $C_P>0$ is the Poincaré constant on unit blocks (A.15).

We now estimate the remaining integral. Assume without loss of generality that . Then, by the triangle inequality and the fundamental theorem of calculus,

Hence, by the Cauchy–Schwarz inequality, the bound on the drift (4.18) and the bound on $\nabla g_K$ (5.12), we have the following estimate: for any $\zeta > 0$,

Thus, by arguing as in Sect. 5.8.1, one can show the following estimate: let $\zeta < \frac{1}{16C_P}$ and $\varepsilon = 1 - 8\zeta C_P > 0$. Then, provided $\eta < \frac{1}{392 C_P}$ and $\beta > 1$,

$$\begin{aligned} {\mathbb {E}}\Psi _K^{Q_3}(v)&\geqslant {\mathbb {E}}\Bigg [ -CN^\Xi _K + \Big ( \frac{1-\varepsilon }{2} - \frac{2\zeta C_P}{\beta ^3} - \frac{2\zeta C_P}{\beta ^3} \Big ) \int _{{\mathbb {T}}_N}{\mathcal {V}}_\beta (V_K+g_K) dx \\&\quad \quad \quad + \Big ( \frac{1-\varepsilon }{2} - 4\zeta C_P - 4\zeta C_P \Big ) \int _{{\mathbb {T}}_N}\int _0^Kr_k^2 dk dx \\&\quad \quad \quad - (4\zeta C_P + 4\zeta C_P) \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx \Bigg ] \\&\geqslant -CN^3 - 8\zeta C_P \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx. \end{aligned}$$

Applying Proposition 4.7 with Hamiltonian ${\mathcal {H}}_{\beta ,N,K}^{Q_3,\sigma }(\phi _K)$, with total energy cutoff suppressed (see Remark 4.8), yields

$$\begin{aligned} F_{\beta ,N,K}^\sigma - \log {\mathbb {E}}_N e^{-{\mathcal {H}}_{\beta ,N,K}^{Q_3}(\phi _K)} \geqslant - CN^3 + \Big ( \frac{1}{2} - 8\zeta C_P \Big ) \int _{{\mathbb {T}}_N}|\nabla {\tilde{g}}_K|^2 dx \geqslant -CN^3. \end{aligned}$$

This estimate is uniform over all $2^{N^3}$ choices of $\sigma$, hence establishing (5.89). $\quad \square$

6 Chessboard Estimates

In this section we prove Proposition 3.6 using the chessboard estimates of Proposition 6.5 and the estimates obtained in Sect. 5. In addition, we establish that ${\nu _{\beta ,N}}$ is reflection positive.

6.1 Reflection positivity of ${\nu _{\beta ,N}}$

We begin by defining reflection positivity for general measures on spaces of distributions following [Shl86] and [GJ87].

For any $a \in \{0, \dots , N-1\}$ and $\{i,j,k\} = \{ 1,2,3 \}$, let

$$\begin{aligned} {\mathcal {R}}_{\Pi _{a,i}}(x) = (2a-x_i)e_i + e_j + e_k \end{aligned}$$

where $x = x_i e_i + x_j e_j + x_k e_j \in {\mathbb {T}}_N$ and addition is understood modulo N. Define

$$\begin{aligned} \Pi _{a,i} = \{ x \in {\mathbb {T}}_N: {\mathcal {R}}_{\Pi _{a,i}}(x)=x\}. \end{aligned}$$

(6.1)

Note that for any $x \in \Pi _{a,i}$, $x_i = a \text { or } a+\frac{N}{2}$. We say that ${\mathcal {R}}_{\Pi _{a,i}}$ is the reflection map across the hyperplane $\Pi _{a,i}$.

Fix such a hyperplane $\Pi$. It separates ${\mathbb {T}}_N= {\mathbb {T}}_N^+ \sqcup \Pi \sqcup {\mathbb {T}}_N^-$ such that ${\mathbb {T}}_N^+ = {\mathcal {R}}_\Pi {\mathbb {T}}_N^-$. For any $f \in C^\infty ({\mathbb {T}}_N)$, we say f is ${\mathbb {T}}_N^+$-measurable if $\text {supp} f \subset {\mathbb {T}}_N^+$. The reflection of f in $\Pi$ is defined pointwise by ${\mathcal {R}}_\Pi f(x) = f({\mathcal {R}}_\Pi x)$. For any $\phi \in S'({\mathbb {T}}_N)$, we say that $\phi$ is ${\mathbb {T}}_N^+$-measurable if, for any $f \in C^\infty ({\mathbb {T}}_N)$ such that $\mathrm{supp} f \subset ({{\mathbb {T}}_N^+})^c$, $\phi (f) = 0$, where $\phi (f)$ denotes the duality pairing between $S'({\mathbb {T}}_N)$ and $C^\infty ({\mathbb {T}}_N)$. For any such $\phi$, we define ${\mathcal {R}}_\Pi \phi$ pointwise by ${\mathcal {R}}_\Pi \phi (f) = \phi ({\mathcal {R}}_\Pi f)$.

Let $\nu$ be a probability measure on $S'({\mathbb {T}}_N)$. We say that $F \in L^2(\nu )$ is ${\mathbb {T}}_N^+$-measurable if it is measurable with respect to the $\sigma$-algebra generated by the set of $\phi \in S'({\mathbb {T}}_N)$ that are ${\mathbb {T}}_N^+$-measurable. For any such F, we define ${\mathcal {R}}_\Pi F$ pointwise by ${\mathcal {R}}_\Pi F(\phi ) = F({\mathcal {R}}_\Pi \phi )$.

The measure $\nu$ on $S'({\mathbb {T}}_N)$ is called reflection positive if, for any hyperplane $\Pi$ of the form (6.1),

$$\begin{aligned} \int _{S'({\mathbb {T}}_N)} F(\phi ) \cdot {\mathcal {R}}_\Pi F(\phi ) d\nu (\phi ) \geqslant 0 \end{aligned}$$

for all $F \in L^2(\nu )$ that are ${\mathbb {T}}_N^+$-measurable.

Proposition 6.1

The measure ${\nu _{\beta ,N}}$ is reflection positive.

6.1.1 Proof of Proposition 6.1

In general, Fourier approximations to ${\nu _{\beta ,N}}$ (such as $\nu _{\beta ,N,K}$) are not reflection positive. Instead, we prove Proposition 6.1 by considering lattice approximations to ${\nu _{\beta ,N}}$ for which reflection positivity is straightforward to show.

Let ${\mathbb {T}}_N^{\varepsilon } = (\varepsilon {\mathbb {Z}}/ N {\mathbb {Z}})^3$ be the discrete torus of sidelength N and lattice spacing $\varepsilon > 0$. In order to use discrete Fourier analysis, we assume that $\varepsilon ^{-1} \in {\mathbb {N}}$. Note that any hyperplane $\Pi$ of the form (6.1) is a subset of ${\mathbb {T}}_N^\varepsilon$.

For any $\varphi \in ({\mathbb {R}})^{{\mathbb {T}}_N^\varepsilon }$, define the lattice Laplacian

$$\begin{aligned} \Delta ^\varepsilon \varphi (x) = \frac{1}{\varepsilon ^2} \sum _{\begin{array}{c} y \in {\mathbb {T}}_N^\varepsilon \\ |x-y| = \varepsilon \end{array}} (\varphi (y) - \varphi (x)). \end{aligned}$$

Let ${\tilde{\mu }}_{N,\varepsilon }$ be the Gaussian measure on ${\mathbb {R}}^{{\mathbb {T}}_N^\varepsilon }$ with density

$$\begin{aligned} d{\tilde{\mu }}_{N,\varepsilon }(\varphi ) \propto \exp \Big (-\frac{\varepsilon ^3}{2} \sum _{x \in {\mathbb {T}}_N^\varepsilon } \varphi (x) \cdot (-\Delta ^\varepsilon + \eta )\varphi (x) \Big ) \prod _{x \in {\mathbb {T}}_N^\varepsilon } d\varphi (x) \end{aligned}$$

where $d{\phi }(x)$ is Lebesgue measure.

A natural lattice approximation to ${\nu _{\beta ,N}}$ is given by the probability measure ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ with density proportional to

$$\begin{aligned} d{\tilde{\nu }}_{\beta ,N,\varepsilon }(\varphi ) \propto e^{-{\tilde{{\mathcal {H}}}}_{\beta ,N,\varepsilon }(\varphi )} d\tilde{\mu }_{N,\varepsilon }(\varphi ) \end{aligned}$$

where

$$\begin{aligned} {\tilde{{\mathcal {H}}}}_{\beta ,N,\varepsilon }(\varphi ) = \varepsilon ^3 \sum _{x \in {\mathbb {T}}_n^\varepsilon } {\mathcal {V}}_\beta (\varphi (x)) - \Big (\frac{\eta }{2} + \frac{1}{2} \delta m^2(\varepsilon , \eta ) \Big ) \varphi (x)^2 \end{aligned}$$

where $\frac{1}{2} \delta m^2(\varepsilon ,\eta )$ is a renormalisation constant that diverges as $\varepsilon \rightarrow 0$ (see Proposition 6.19). Note two things: first, the renormalisation constant is chosen dependent on $\eta$ for technical convenience. Second, no energy renormalisation is included since we are only interested in convergence of measures.

Remark 6.2

By embedding ${\mathbb {R}}^{{\mathbb {T}}_N^\varepsilon }$ into $S'({\mathbb {T}}_N)$, we can define reflection positivity for lattice measures. We choose this embedding so that the pushforward of ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ is automatically reflection positive, but other choices are possible.

For any $\varphi \in {\mathbb {R}}^{{\mathbb {T}}_N^\varepsilon }$, we write $\mathrm {ext}^\varepsilon \varphi$ for its unique extension to a trigonometric polynomial on ${\mathbb {T}}_N$ of degree less than $\varepsilon ^{-1}$ that coincides with $\varphi$ on lattice points (i.e. in ${\mathbb {T}}_N^\varepsilon$). Precisely,

$$\begin{aligned} \mathrm {ext}^\varepsilon (\varphi )(x) = \frac{\varepsilon ^3}{N^3} \sum _{n} \sum _{ y \in {\mathbb {T}}_N^\varepsilon } e_n(y-x) \varphi (y) \end{aligned}$$

where the sum ranges over all $n=(a_1,a_2,a_3) \in (N^{-1}{\mathbb {Z}})^3$ such that $|a_i| \leqslant \varepsilon ^{-1}$, and we recall $e_n(x) = e^{2\pi i n \cdot x}$.

Lemma 6.3

Let $\varepsilon > 0$ such that $\varepsilon ^{-1} \in {\mathbb {N}}$. Denote by $\mathrm {ext}^\varepsilon _* {\tilde{\nu }}_{\beta ,N,\varepsilon }$ the pushforward of ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ by the map $\mathrm {ext}^\varepsilon$. Then, the measure $\mathrm {ext}^\varepsilon _* {\tilde{\nu }}_{\beta ,N,\varepsilon }$ is reflection positive.

Proof

Fix a hyperplane $\Pi$ of the form (6.1) and recall that $\Pi$ separates ${\mathbb {T}}_N={\mathbb {T}}_N^+ \sqcup \Pi \sqcup {\mathbb {T}}_N^-$. Write ${\mathbb {T}}_{N,\varepsilon }^+ = {\mathbb {T}}_N^+ \cap {\mathbb {T}}_N^\varepsilon$.

Since the measure ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ is reflection positive on the lattice by [Shl86, Theorem 2.1], the following estimate holds: let $F^\varepsilon \in L^2({\tilde{\nu }}_{\beta ,N,\varepsilon })$ be ${\mathbb {T}}_{N,\varepsilon }^+$-measurable - i.e. $F^\varepsilon (\varphi )$ depends only on $\varphi (x)$ for $x \in {\mathbb {T}}_{N,\varepsilon }^+$. Then,

$$\begin{aligned} \int F^\varepsilon (\varphi ) \cdot {\mathcal {R}}_\Pi F^\varepsilon (\varphi ) d{\tilde{\nu }}_{\beta ,N,\varepsilon }(\varphi ) \geqslant 0. \end{aligned}$$

(6.2)

Let $F \in L^2(\mathrm {ext}^\varepsilon _*{\tilde{\nu }}_{\beta ,N,\varepsilon })$ be ${\mathbb {T}}_N^+$-measurable. Then, $F \circ \mathrm {ext}^\varepsilon \in L^2({\tilde{\nu }}_{\beta ,N,\varepsilon })$ is ${\mathbb {T}}_{N,\varepsilon }^+$-measurable. Using that $\mathrm {ext}^\varepsilon$ and ${\mathcal {R}}_\Pi$ (the reflection across $\Pi$) commute,

$$\begin{aligned} \int F(\phi ) \cdot {\mathcal {R}}_\Pi F(\phi ) d \mathrm {ext}^\varepsilon _* {\tilde{\nu }}_{\beta ,N,\varepsilon }(\phi )&= \int (F \circ \mathrm {ext}^\varepsilon )(\varphi ) \cdot (F \circ {\mathcal {R}}_\Pi \circ \mathrm {ext}^\varepsilon ) (\varphi ) d{\tilde{\nu }}_{\beta ,N,\varepsilon }(\varphi ) \\&= \int (F\circ \mathrm {ext}^\varepsilon ) (\varphi ) \cdot (F \circ \mathrm {ext}^\varepsilon ) ({\mathcal {R}}_\Pi \varphi ) d{\tilde{\nu }}_{\beta ,N,\varepsilon }(\varphi ) \\&\geqslant 0 \end{aligned}$$

where the last inequality is by (6.2). Hence, $\mathrm {ext}^\varepsilon _*{\tilde{\nu }}_{\beta ,N,\varepsilon }$ is reflection positive. $\quad \square$

Proposition 6.4

There exist constants $\frac{1}{2} \delta m^2(\bullet ,\eta )$ such that $\mathrm {ext}^\varepsilon _*\nu _{\beta ,N,\varepsilon } \rightarrow {\nu _{\beta ,N}}$ weakly as $\varepsilon \rightarrow \infty$.

Proof

The existence of a weak limit of $\mathrm {ext}^\varepsilon _*{\tilde{\nu }}_{\beta ,N,\varepsilon }$ as $\varepsilon \rightarrow 0$ was first established in [Par75]. The fact the lattice approximations and the Fourier approximations (i.e. $\nu _{\beta ,N,K}$) yield the same limit as the cutoff is removed is not straightforward in 3D because of the mutual singularity of ${\nu _{\beta ,N}}$ and $\mu _N$ [BG20]. Previous approaches have relied on Borel summation techniques to show that the correlation functions agree with (resummed) perturbation theory [MS77].

In Sect. 6.4 we give an alternative proof using stochastic quantisation techniques. The key idea is to view ${\nu _{\beta ,N}}$ as the unique invariant measure for a singular stochastic PDE with a local solution theory that is robust under different approximations. This allows us to show directly that $\mathrm {ext}^\varepsilon _*{\tilde{\nu }}_{\beta ,N,\varepsilon }$ converges weakly to ${\nu _{\beta ,N}}$ and avoids the use of Borel summation and perturbation theory. The strategy is explained in further detail at the beginning of that section. $\quad \square$

Proof of Proposition 6.1 assuming Proposition 6.4

Proposition 6.1 is a direct consequence of Lemma 6.3 and Proposition 6.4 since reflection positivity is preserved under weak limits. $\quad \square$

6.2 Chessboard estimates for ${\nu _{\beta ,N}}$

Let $B \subset {{\mathbb {B}}_N}$ be either a unit block or a pair of nearest-neighbour blocks. Recall the natural identification of B with the subset of ${\mathbb {T}}_N$ given by the union of blocks in B. ${\mathbb {T}}_N$ can be written as a disjoint union of translates of B. Let ${\mathbb {B}}_N^B$ be the set of these translates; its elements are also identified with subsets of ${\mathbb {T}}_N$. Note that if , then ${\mathbb {B}}_N^B = {{\mathbb {B}}_N}$.

We say that $f \in C^\infty ({\mathbb {T}}_N)$ is B-measurable if $\text {supp} f \subset B$ and $\text {supp} f \cap \partial B = \emptyset$. We say that $\phi \in S'({\mathbb {T}}_N)$ is B-measurable if $\phi (f) = 0$ for every $f \in C^\infty ({\mathbb {T}}_N)$ unless f is B-measurable. We say that $F \in L^2({\nu _{\beta ,N}})$ is B-measurable if it is measurable with respect to the $\sigma$-algebra generated by $\phi \in S'({\mathbb {T}}_N)$ that are B-measurable.

Proposition 6.5

Let $N \in 4{\mathbb {N}}$. Let $\{ F_{{\tilde{B}}} : {\tilde{B}} \in {\mathbb {B}}_N^B \}$ be a given set of $L^2({\nu _{\beta ,N}})$-functions such that each $F_{{\tilde{B}}}$ is ${\tilde{B}}$-measurable.

Fix ${\tilde{B}} \in {\mathbb {B}}_N^B$ and define an associated set of $L^2({\nu _{\beta ,N}})$-functions $\{ F_{{\tilde{B}}, B'} : B' \in {\mathbb {B}}_N^B \}$ by the conditions: $F_{{\tilde{B}}, {\tilde{B}}} = F_{{\tilde{B}}}$; and, for any $B', B'' \in {\mathbb {B}}_N^B$ such that $B'$ and $B''$ share a common face,

$$\begin{aligned} F_{{\tilde{B}}, B'} = {\mathcal {R}}_\Pi F_{{\tilde{B}}, B''} \end{aligned}$$

where $\Pi$ is the unique hyperplane of the form (6.1) containing the shared face between $B'$ and $B''$.

Then,

$$\begin{aligned} \Big | \Big \langle \prod _{{\tilde{B}} \in {\mathbb {B}}_N^B} F_{{\tilde{B}}} \Big \rangle _{\beta ,N} \Big | \leqslant \prod _{{\tilde{B}} \in {\mathbb {B}}_N^B} \Big | \Big \langle \prod _{B' \in {\mathbb {B}}_N^B} F_{{\tilde{B}}, B'} \Big \rangle _{\beta ,N} \Big |^ \frac{|B|}{N^3}. \end{aligned}$$

Proof

This is a consequence of the reflection positivity of ${\nu _{\beta ,N}}$. The condition $N \in 4{\mathbb {N}}$ guarantees $F_{{\tilde{B}}, B'}$ is well-defined. See [Shl86, Theorem 2.2]. $\quad \square$

6.3 Proof of Proposition 3.6

In order to be able to apply Proposition 6.5 to the random variables $Q_i$ of Proposition 3.6, we need the following lemma.

Lemma 6.6

Let $N \in {\mathbb {N}}$ and $\beta > 0$. Then, for any , is -measurable.

In addition, for any nearest neighbours , is -measurable.

Proof

The fact that follows from estimates obtained in Proposition 5.23. The and measurability of these observables comes from taking approximations to indicators which are supported on blocks (e.g. using some appropriate regularisation of the distance function) and estimates obtained in Proposition 5.23. $\quad \square$

Proof of Proposition 3.6

Let $B_1, B_2 \subset {{\mathbb {B}}_N}$ and $B_3$ be a set of unordered pairs of nearest neighbour blocks in ${{\mathbb {B}}_N}$. Then,

(6.3)

where $\cosh Q_i(B_i)$ is defined in (3.7) and the sum is over all partitions $B_1^+ \sqcup B_1^- = B_1$ and $B_2^+\sqcup B_2^- = B_2$.

It suffices to prove that there exists ${\tilde{C}}_Q>0$ such that, for any $B_1^\pm , B_2^\pm$ and $B_3$ as above,

(6.4)

Then, taking expectations in (6.3) and using (6.4)

which yields Proposition 3.6 with $C_Q = {\tilde{C}}_Q$.

To prove (6.4), first fix $B_1^\pm$ and $B_2^\pm$. Then, by Hölder’s inequality,

(6.5)

Let $i=1,2$. Without loss of generality, we use Proposition 6.5 to estimate

Define if and 1 otherwise. For each , we generate the family of functions as in Proposition 6.5. Note that for such that and are nearest-neighbours,

where ${\mathcal {R}}$ is the reflection across the unique hyperplane containing the shared face of and . Thus, we have for every and . If , we have for every .

Lemma 6.6 ensures that is -measurable for every . Hence, by Proposition 6.5, we obtain

Therefore, by Proposition 5.1, there exists $C_Q'>0$ such that, for all $\beta$ sufficiently large,

(6.6)

For the remaining term involving $Q_3$, partition $B_3 = \bigcup _{k=1}^6 B_3^{(k)}$ such that each $B_3^{(k)}$ is a set of disjoint pairs of nearest neighbour blocks, all with same orientation. Then, by Hölder’s inequality,

(6.7)

Assuming that we have established that there exists $C_Q'>0$ such that

for every $k \in \{1,\dots ,6\}$, then (6.7) yields

Hence, without loss of generality, we may assume $B_3$ is a set of disjoint pairs of nearest neighbour blocks, all of the same orientation.

Define for any and 1 otherwise. Note that for any two pairs of nearest-neighbour blocks, ,

where ${\mathcal {R}}$ is the reflection across the unique hyperplane containing the shared face of . Thus, for any and $B$, we have . If $B \not \in B_3$, then we have $F_{B,B'}=1$ for all $B' \in {\mathbb {B}}_N^B$.

Lemma 6.6 ensures that is -measurable. Thus, applying Propositions 6.5 and 5.1 , there exists $C_Q'>0$ such that, for all $\beta$ sufficiently large,

(6.8)

Inserting (6.6) and (6.8) into (6.5), and taking into account (6.7), yields (6.4) with ${\tilde{C}}_Q = \frac{C_Q'}{15}$, thereby finishing the proof. $\quad \square$

6.4 Equivalence of the lattice and Fourier cutoffs

This section is devoted to a proof of Proposition 6.4 using stochastic quantisation techniques. In Sect. 6.4.1, we give a rigorous interpretation to (1.3) via the change of variables (6.14). Subsequently, in Sect. 6.4.2, we establish that ${\nu _{\beta ,N}}$ is the unique invariant measure of (1.3), see Proposition 6.18. In Sect. 6.4.3, we first establish that local solutions of spectral Galerkin and lattice approximations to (1.3) converge to the same limit (see Propositions 6.13 and 6.19 ); these approximations admit unique invariant measures given by $\nu _{\beta ,N,K}$ and ${\tilde{\nu }}_{\beta ,N,\varepsilon }$, respectively. Then, using the global existence of solutions and uniqueness of the invariant measure of (1.3), we show that both of these measures converge to ${\nu _{\beta ,N}}$ as the cutoffs are removed.

6.4.1 Giving a meaning to (1.3)

Let $\xi$ be space-time white noise on ${\mathbb {T}}_N$ defined on a probability space . This means that $\xi$ is a Gaussian random distribution on $\Omega$ satisfying

where $\Phi ,\Psi \in C^\infty ({\mathbb {R}}_+\times {\mathbb {T}}_N)$ and denotes expectation with respect to . We use the colour blue here to distinguish between the space random processes defined in Sect. 4 and the space-time random processes that we consider here.

We interpret (1.3) as the limit of renormalised approximations. For every $K \in (0,\infty )$, the Glauber dynamics of $\nu _{\beta ,N,K}$ is given by the stochastic PDE

(6.9)

Above, $\rho _K$ is as in Sect. 2 and we recall $\rho _K^2 \ne \rho _K$; is defined in (2.1); and , where is defined in (4.4).

Remark 6.7

Recall that the Glauber dynamics for the measure $\nu$ with formal density $d\nu (\phi ) \propto e^{-{\mathcal {H}}(\phi )}\prod _{x\in {\mathbb {T}}_N} d\phi (x)$ is given by the (overdamped) Langevin equation

$$\begin{aligned} \partial _t \Phi (t) = \partial _\phi {\mathcal {H}}(\Phi (t)) + \sqrt{2} \xi \end{aligned}$$

where $\partial _\phi {\mathcal {H}}$ denotes the functional derivative of ${\mathcal {H}}$.

For fixed K, the (almost sure) global existence and uniqueness of mild solutions to (6.9) is standard (see e.g. [DPZ88, Section III]). Moreover, $\nu _{\beta ,N,K}$ is its unique invariant measure (see [Zab89, Theorem 2]). The approximations (6.9), which we call spectral Galerkin approximations, are natural in our context since ${\nu _{\beta ,N}}$ is constructed as the weak limit of $\nu _{\beta ,N,K}$ as $K \rightarrow \infty$.

The difficulty in obtaining a local well-posedness theory that is stable in the limit $K \rightarrow \infty$ lies in the roughness of the white noise $\xi$. The key idea is to exploit that the small-scale behaviour of solutions to (6.9) is governed by the Ornstein-Uhlenbeck process

This allows us to obtain an expansion of $\Phi _K$ in terms of explicit (renormalised) multilinear functions of , which give a more detailed description of the small-scale behaviour of $\Phi _K$, plus a more regular remainder term. Given the regularities of these explicit stochastic terms, the local solution theory then follows from deterministic arguments.

Remark 6.8

We are only concerned with the limit $K \rightarrow \infty$ in (6.9). We do not try to make sense of the joint $K,N \rightarrow \infty$ limit.

We use the paracontrolled distribution approach of [MW17], which is modification of the framework of [CC18] (both influenced by the seminal work of [GIP15]). In this approach, the expansion of $\Phi _K$ is given by an ansatz, see (6.10), that has similarities to the change of variables encountered in Sect. 5.4.1. See Remark 6.10. There are also related approaches via regularity structures [Hai14, Hai16, MW18] and renormalisation group [Kup16], but we do not discuss them further.

For every $K \in (0,\infty )$, define

We recall that the colour blue is used to distinguish between the above space-time diagrams and the space diagrams of Sect. 4.1.1.

For any $T>0$, the vector is space-time stationary and almost surely an element of the Banach space

$$\begin{aligned} {\mathcal {X}}_T&= C([0,T]; {\mathcal {C}}^{-\frac{1}{2} - \kappa })\times C([0,T]; {\mathcal {C}}^{-1-\kappa }) \\&\quad \times \Big ( C([0,T]; {\mathcal {C}}^{\frac{1}{2} - \kappa }) \cap C^\frac{1}{8}([0,T]; {\mathcal {C}}^{\frac{1}{4} - \kappa }) \Big ) \\&\quad \times C([0,T]; {\mathcal {C}}^{-\kappa })\times C([0,T]; {\mathcal {C}}^{-\kappa }) \times C([0,T]; {\mathcal {C}}^{-\frac{1}{2} -\kappa }) \end{aligned}$$

where the norm on ${\mathcal {X}}_T$ is given by the maximum of the norms on the components. Above, for any $s \in {\mathbb {R}}$, $C([0,T];{\mathcal {C}}^s)$ consists of continuous functions $\Phi :[0,T] \rightarrow {\mathcal {C}}^s$ and is a Banach space under the norm $\sup _{t\in [0,T]}\Vert \cdot \Vert _{{\mathcal {C}}^s}$. In addition, for any $\alpha \in (0,1)$, $C^\alpha ([0,T];{\mathcal {C}}^s)$ consists of $\alpha$-Hölder continuous functions $\Phi : [0,T] \rightarrow {\mathcal {C}}^s$ and is a Banach space under the norm $\Vert \cdot \Vert _{C([0,T];{\mathcal {C}}^s)} + | \cdot |_{\alpha , T}$ where

$$\begin{aligned} | \Phi |_{\alpha ,T} = \sup _{0<s<t<T} \frac{\Vert \Phi (t) - \Phi (s)\Vert _{{\mathcal {C}}^s}}{|t-s|^\alpha }. \end{aligned}$$

Proposition 6.9

There exists a stochastic process such that, for every $T > 0$, almost surely and

Proof

The proof follows from [CC18, Section 4] (see also [MWX17] and [Hai14, Section 10]). The only subtlety is to check that the renormalisation constants and , which were determined by the field theory ${\nu _{\beta ,N}}$, are sufficient to renormalise the space-time diagrams appearing in the analysis of the SPDE. Precisely, it suffices to show and for every $(t,x) \in {\mathbb {R}}_+\times {\mathbb {T}}_N$.

There exists a set of complex Brownian motions $\{ W^n(\bullet ) \}_{n \in (N^{-1}{\mathbb {Z}})^3}$ defined on , independent modulo the condition $W^n(\bullet ) = \overline{W^{-n}(\bullet )}$, such that

$$\begin{aligned} \xi (\phi )&= \frac{1}{N^3} \sum _{n \in (N^{-1}{\mathbb {Z}})^3} \int _{\mathbb {R}}{\mathcal {F}}(\phi )(t,n) N^\frac{3}{2} dW^n(t) \end{aligned}$$

for every $\phi \in L^2({\mathbb {R}}\times {\mathbb {T}}_N)$.

For $t \geqslant 0$ and $n \in (N^{-1}{\mathbb {Z}})^3$, let $H(t,n) = e^{-t \langle n \rangle ^2}$ be the (spatial) Fourier transform of the heat kernel associated to $(\partial _t - \Delta + \eta )$. For any $K>0$, define $H_K(t,n) = \rho _K (n) H(t,n)$. We extend both kernels to $t \in {\mathbb {R}}$ by setting $H(t,\cdot ) = H_K(t,\cdot ) = 0$ for any $t < 0$. Then

By Parseval’s theorem and Itô’s isometry,

for all $(t,x) \in {\mathbb {R}}_+\times {\mathbb {T}}_N$. With this observation the convergence of , and follows from mild adaptations of [CC18, Section 4].

For the remaining two diagrams, one can show from arguments in [CC18, Section 4] that

converge to well-defined space-time distributions.

Writing

we have, by Parseval’s theorem and Itô’s isometry,

By symmetry,

thereby completing the proof. $\quad \square$

We return now to the solution theory for (1.3)/(6.9). Fix $K \in (0,\infty )$. Using the change of variables

(6.10)

we say that $\Phi _K$ is a mild solution of (6.9) with initial data $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$ if $(\Upsilon _K, \Theta _K)$ is a mild solution to the system of equations

(6.11)

where

with initial data .

We split ,

where and are commutator terms defined through the manipulations

(6.12)

The precise choice of the splitting of into $\Upsilon _K$ and $\Theta _K$ is explained in detail in [MW17, Introduction]. For our purposes, it suffices to note that $\Upsilon _K$ captures the small-scale behaviour of this difference. On the other hand, $\Theta _K$ captures the large-scale behaviour: the term $G_K^2$ contains a cubic damping term in $\Theta _K$ (i.e. with a good sign). Finally, we note that there is a redundancy in the specification of initial condition: any choice such that is sufficient. Our choice is informed by Remark 1.3 in [MW17].

Remark 6.10

Rewriting (6.10) as

we note the similarity between the change of variables for the stochastic PDE given above and for the field theory in (5.21).

Formally taking $K \rightarrow \infty$ in (6.11) leads us to the following system:

(6.13)

where

and $G^{1,a}$ and $G^{1,b}$ are commutator terms defined analogously as in (6.12).

For every $T>0$, define the Banach space

$$\begin{aligned} {\mathcal {Y}}_T&= \Big [ C([0,T];{\mathcal {C}}^{-\frac{3}{5}}) \cap C((0,T]; {\mathcal {C}}^{\frac{1}{2} + 2\kappa }) \cap C^\frac{1}{8} ( (0,T]; L^\infty ) \Big ] \\&\quad \times \Big [ C([0,T]; {\mathcal {C}}^{-\frac{3}{5}}) \cap C((0,T]; {\mathcal {C}}^{1+2\kappa }) \cap C^\frac{1}{8} ( (0,T]; L^\infty ) \Big ] \end{aligned}$$

equipped with the norm

$$\begin{aligned}&\Vert (\Upsilon , \Theta ) \Vert _{{\mathcal {Y}}_T} \\&\quad = \max \Bigg \{ \sup _{0 \leqslant t \leqslant T} \Vert \Upsilon (t) \Vert _{{\mathcal {C}}^{-\frac{3}{5}}}, \sup _{0< t \leqslant T}t^\frac{3}{5} \Vert \Upsilon (t) \Vert _{{\mathcal {C}}^{\frac{1}{2} + 2\kappa }}, \sup _{0< s< t \leqslant T} s^\frac{1}{2} \frac{\Vert \Upsilon (t) - \Upsilon (s) \Vert _{L^\infty }}{|t-s|^\frac{1}{8}}, \\&\quad \quad \sup _{0 \leqslant t \leqslant T} \Vert \Theta (t) \Vert _{{\mathcal {C}}^{-\frac{3}{5}}}, \sup _{0< t \leqslant T} t^\frac{17}{20} \Vert \Theta (t) \Vert _{{\mathcal {C}}^{1+2\kappa }}, \sup _{0< s < t \leqslant T} s^\frac{1}{2} \frac{\Vert \Theta (t) - \Theta (s) \Vert _{L^\infty }}{|t-s|^\frac{1}{8}} \Bigg \}. \end{aligned}$$

Remark 6.11

The choice of exponents in function spaces in ${\mathcal {Y}}_T$, as well as the choice of exponents in the blow-up at $t=0$ in $\Vert \cdot \Vert _{{\mathcal {Y}}_T}$, corresponds to the one made in [MW17]. It is arbitrary to an extent: it depends on the choice of initial condition, which must have Besov-Hölder regularity strictly better than $-\frac{2}{3}$.

The local well-posedness of (6.13) follows from entirely deterministic arguments, so we state it with replaced by any deterministic .

Proposition 6.12

Let for any $T_0>0$, and let $(\Upsilon _0,\Theta _0) \in {\mathcal {C}}^{-\frac{3}{5}}\times {\mathcal {C}}^{-\frac{3}{5}}$. Then, there exists such that there is a unique mild solution $(\Upsilon , \Theta ) \in {\mathcal {Y}}_T$ to (6.13) with initial data $(\Upsilon _0,\Theta _0)$.

In addition, let such that for some $R>0$, and let $(\Upsilon _0^1,\Theta _0^1),(\Upsilon _0^2,\Theta _0^2) \in {\mathcal {C}}^{-\frac{3}{5}} \times {\mathcal {C}}^{-\frac{3}{5}}$. Let the respective solutions to (6.13) be $(\Upsilon ^1,\Theta ^1) \in {\mathcal {Y}}_{T_1}$ and $(\Upsilon ^2,\Theta ^2) \in {\mathcal {Y}}_{T_2}$ and define $T=\min (T_1,T_2)$. Then there exists $C=C(R)>0$ such that

Proof

Proposition 6.12 is proven in Theorem 2.1 [MW17] (see also Theorem 3.1 [CC18]) by showing that the mild solution map

is a contraction in the ball

$$\begin{aligned} {\mathcal {Y}}_{T,M} = \Big \{ ({\tilde{\Upsilon }}, {\tilde{\Theta }}) \in {\mathcal {Y}}_T : \Vert ({\tilde{\Upsilon }}, {\tilde{\Theta }}) \Vert _{{\mathcal {Y}}_T} \leqslant M \Big \} \end{aligned}$$

provided that T is taken sufficiently small and M is taken sufficiently large (both depending on the norm of the initial data and of ). $\quad \square$

We say that $\Phi \in C([0,T];{\mathcal {C}}^{-\frac{1}{2}-\kappa })$ is a mild solution to (1.3) with initial data $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$ if

(6.14)

where $(\Upsilon ,\Theta ) \in {\mathcal {Y}}_T$ is a solution to (6.13) with as in Proposition 6.9 and initial data .

Proposition 6.13

For any $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$, let $\Phi \in C([0,T];{\mathcal {C}}^{-\frac{1}{2}-\kappa })$ be the unique solution of (1.3) with initial data $\phi _0$ up to time $T>0$. In addition, for any $K \in (0,\infty )$, let $\Phi _K \in C({\mathbb {R}}_+;{\mathcal {C}}^{-\frac{1}{2}-\kappa })$ be the unique global solution of (6.9) with initial data $\rho _K\phi _0$.

Then,

Proof

It suffices to show convergence of $(\Upsilon _K,\Theta _K)$ to $(\Upsilon ,\Theta )$ as $K \rightarrow \infty$. This follows from Proposition 6.9 and mild adaptations of arguments in [MW17, Section 2]. $\quad \square$

Proposition 6.13 implies that $\Phi _K \rightarrow \Phi$ in probability in $C([0,T];{\mathcal {C}}^{-\frac{1}{2}-\kappa })$. Local-in-time convergence is not sufficient for our purposes.

The following proposition establishing global well-posedness of (1.3).

Proposition 6.14

For every $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$ let $\Phi \in C([0,T^*);{\mathcal {C}}^{-\frac{1}{2}-\kappa })$ be the unique solution to (1.3) with initial condition $\phi _0$ and where $T^*>0$ is the maximal time of existence. Then $T^* = \infty$ almost surely.

Proof

Proposition 6.14 is a consequence of a strong a priori bound on solutions to (6.13) established in [MW17, Theorem 1.1]. $\quad \square$

An immediate corollary of Proposition 6.14 is a global-in-time convergence result sufficient for our purposes.

Corollary 6.15

For every $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$, let $\Phi \in C({\mathbb {R}}_+;{\mathcal {C}}^{-\frac{1}{2}-\kappa })$ be the unique global solution to (1.3) with initial condition $\phi _0$. For every $K \in (0,\infty )$, let $\Phi _K \in C({\mathbb {R}}_+;{\mathcal {C}}^{-\frac{1}{2}-\kappa })$ be the unique global solution to (6.9) with initial condition $\rho _K \phi _0$.

For every $T>0$,

Remark 6.16

The infinite constant in (1.3) represents the renormalisation constants of the approximating equation (6.9) going to infinity as $K \rightarrow \infty$. Note that there is a one-parameter family of distinct nontrivial “solutions” to (1.3) corresponding to taking finite shifts of the renormalisation constants. However, the use of in the change of variables (6.14) fixes the precise solution.

6.4.2 ${\nu _{\beta ,N}}$ is the unique invariant measure of (6.14)

Denote by $B_b({\mathcal {C}}^{-\frac{1}{2} -\kappa })$ the set of bounded measurable functions on ${\mathcal {C}}^{-\frac{1}{2} -\kappa }$ and by $C_b({\mathcal {C}}^{-\frac{1}{2} - \kappa }) \subset B_b({\mathcal {C}}^{-\frac{1}{2}-\kappa })$ the set of bounded continuous functions on ${\mathcal {C}}^{-\frac{1}{2} - \kappa }$.

Let $\Phi (\cdot ;\cdot )$ be the solution map to (1.3): for $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2} - \kappa }$ and $t \in {\mathbb {R}}_+$, $\Phi (t;\phi _0)$ is the solution at time t to (1.3) with initial condition $\phi _0$. For every $t>0$, define ${\mathcal {P}}^{\beta ,N}_t:B_b({\mathcal {C}}^{-\frac{1}{2}-\kappa }) \rightarrow B_b({\mathcal {C}}^{-\frac{1}{2}-\kappa })$ by

for $F \in B_b({\mathcal {C}}^{-\frac{1}{2} - \kappa })$, $\phi _0 \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$, and $t \in {\mathbb {R}}_+$.

Proposition 6.17

The solution $\Phi$ to (1.3) is a Markov process and its transition semigroup $({\mathcal {P}}^{\beta ,N}_t)_{t\geqslant 0}$ satisfies the strong Feller property, i.e. ${\mathcal {P}}_t:B_b({\mathcal {C}}^{-\frac{1}{2}-\kappa }) \rightarrow C_b({\mathcal {C}}^{-\frac{1}{2}-\kappa })$.

Proof

See [HM18b, Theorem 3.2]. $\quad \square$

Proposition 6.18

The measure ${\nu _{\beta ,N}}$ is the unique invariant measure of (1.3).

Proof

By Proposition 2.1 the measures $\nu _{\beta ,N,K}$ converge weakly to ${\nu _{\beta ,N}}$ as $K \rightarrow \infty$. Hence, by Skorokhod’s representation theorem [Bil08, Theorem 25.6] we can assume that there exists a sequence of random variables $\{\phi _K\}_{K \in {\mathbb {N}}} \subset {\mathcal {C}}^{-\frac{1}{2} - \kappa }$ defined on the probability space , independent of the white noise $\xi$, such that $\phi _K \sim \nu _{\beta ,N,K}$ and $\phi _K$ converges almost surely to a random variable $\phi \sim {\nu _{\beta ,N}}$.

For every $K \in (0,\infty )$, recall that the unique invariant measure of (6.9) is $\nu _{\beta ,N,K}$. Let $\Phi _K$ denote the solution to (6.9) with random initial data $\phi _K$. Hence, $\Phi _K(t) \sim \nu _{\beta ,N,K}$ for all $t \in {\mathbb {R}}_+$.

Denote by $\Phi$ the solution to (1.3) with initial condition $\phi$. By Proposition 6.14, $\Phi _K(t)$ converges in distribution to $\Phi (t)$ for every $t \in {\mathbb {R}}$, which implies $\Phi (t) \sim {\nu _{\beta ,N}}$. Thus, ${\nu _{\beta ,N}}$ is an invariant measure of (1.3). As a consequence of the strong Feller property in Proposition 6.17, we obtain that ${\nu _{\beta ,N}}$ is the unique invariant measure of (1.3). $\quad \square$

6.4.3 Proof of Proposition 6.4

The Glauber dynamics of ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ is given by the system of SDEs

$$\begin{aligned} \frac{d}{dt} {\tilde{\Phi }}&= \Delta ^\varepsilon {\tilde{\Phi }} - \frac{4}{\beta }{\tilde{\Phi }}^3 + (4+\delta m^2(\varepsilon ,\eta )){\tilde{\Phi }} + \sqrt{2} \xi _\varepsilon \nonumber \\ {\tilde{\Phi }}(0,\cdot )&= \varphi (\cdot ) \end{aligned}$$

(6.15)

where ${\tilde{\Phi }} : {\mathbb {R}}_+\times {\mathbb {T}}_N^\varepsilon \rightarrow {\mathbb {R}}$, $\varphi \in {\mathbb {R}}^{{\mathbb {T}}_N^\varepsilon }$, and $\xi _\varepsilon$ is the lattice discretisation of $\xi$ given by

Note that the integral above means duality pairing between $\xi (t,\cdot )$ and .

For each $\varepsilon > 0$, the global existence and uniqueness of (6.15), as well as the fact that ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ is its unique invariant measure, is well-known.

The following proposition establishes a global-in-time convergence result for solutions of (6.15) to solutions of (1.3).

Proposition 6.19

For every $\varepsilon > 0$, denote by ${\tilde{\Phi }}^\varepsilon$ the unique global solution to (6.15) with initial data $\varphi _{\varepsilon } \in {\mathbb {R}}^{{\mathbb {T}}_N^\varepsilon }$. In addition, denote by $\Phi$ the unique global solution to (1.3) with initial data $\phi \in {\mathcal {C}}^{-\frac{1}{2}-\kappa }$.

Then, there exists a choice of constants $\delta m^2(\varepsilon ,\eta ) \rightarrow \infty$ as $\varepsilon \rightarrow 0$ such that, for every $T > 0$,

provided that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\Vert \phi - \mathrm {ext}^\varepsilon \varphi _{\varepsilon } \Vert _{{\mathcal {C}}^{-\frac{1}{2}-\kappa }} = 0 \end{aligned}$$

(6.16)

almost surely.

Proof

See [ZZ18b, Theorem 1.1] or [HM18a, Theorem 1.1]. $\quad \square$

The next proposition establishes that the lattice measures are tight.

Proposition 6.20

Let $\delta m^2(\bullet ,\eta )$ be as in Proposition 6.19. Then, $\mathrm {ext}^\varepsilon _*{\tilde{\nu }}_{\beta ,N,\varepsilon }$ converges weakly to a measure $\nu$ as $\varepsilon \rightarrow 0$.

Proof

See [Par75, BFS83, GH18]. $\quad \square$

Proof of Proposition 6.4

For every $\varepsilon >0$, let $\varphi _\varepsilon \sim {\tilde{\nu }}_{\beta ,N,\varepsilon }$ be a random variable on and independent of the white noise $\xi$. By Proposition 6.20 and in light of Skorokhod’s representation theorem [Bil08, Theorem 25.6], we may assume that $\mathrm {ext}^\varepsilon \varphi _\varepsilon$ converges almost surely to $\phi \sim \nu$ as $\varepsilon \rightarrow 0$. Reflection positivity is preserved by weak limits hence, by Lemma 6.3, $\nu$ is reflection positive.

Denote by ${\tilde{\Phi }}^\varepsilon$ the solution to (6.15) with initial data $\varphi _\varepsilon$. Since ${\tilde{\nu }}_{\beta ,N,\varepsilon }$ is the invariant measure of (6.15), ${\tilde{\Phi }}^\varepsilon (t) \sim {\tilde{\nu }}_{\beta ,N,\varepsilon }$ for every $t \in {\mathbb {R}}_+$.

Denote by $\Phi$ the (global-in-time) solution to (1.3) with initial data $\phi$. For every $t>0$, $\mathrm {ext}^\varepsilon {\tilde{\Phi }}^\varepsilon (t) \rightarrow \Phi (t)$ in distribution as $\varepsilon \rightarrow 0$ as a consequence of Proposition 6.19. Hence, $\Phi (t) \sim \nu$ for every $t > 0$. Thus, $\nu$ is an invariant measure of (1.3). By Proposition 6.17 the invariant measure of (1.3) is unique. Therefore, $\nu = {\nu _{\beta ,N}}$. $\quad \square$

7 Decay of Spectral Gap

Proof of Corollary 1.3

The Markov semigroup $({\mathcal {P}}_t^{\beta ,N})_{t \geqslant 0}$ associated to (1.3) is reversible with respect to ${\nu _{\beta ,N}}$ (see [HM18a, Corollary 1.3] or [ZZ18a, Lemma 4.2]). Thus, one can express $\lambda _{\beta ,N}$ as the sharpest constant in the Poincaré inequality

$$\begin{aligned} \lambda _{\beta ,N} = \inf _{F \in D({\mathcal {E}}_{\beta ,N})} \frac{{\mathcal {E}}_{\beta ,N}(F,F)}{\langle F^2 \rangle _{\beta ,N} - \langle F \rangle _{\beta ,N}^2} > 0 \end{aligned}$$

(7.1)

where ${\mathcal {E}}_{\beta ,N}$ is the associated Dirichlet form with domain $D({\mathcal {E}}_{\beta ,N}) \subset L^2({\nu _{\beta ,N}})$. See [ZZ18a, Corollary 1.5].

The proof of Corollary 1.3 amounts to choosing the right test function in (7.1) and then using the explicit expression for ${\mathcal {E}}_{\beta ,N}$ for sufficiently nice functions due to [ZZ18a, Theorem 1.2].

Let $\mathrm {Cyl}$ be the set of $F \in L^2({\nu _{\beta ,N}})$ of the form

$$\begin{aligned} F(\cdot ) = f\Big ( l_1(\cdot ),\dots ,l_m(\cdot ) \Big ) \end{aligned}$$

where $m \in {\mathbb {N}}$, $f \in C^1_b({\mathbb {R}}^m)$, $l_1,\dots ,l_m$ are real trigonometric polynomials, and $l_i(\cdot )$ denotes the ($L^2$) duality pairing between $l_i$ and elements in ${\mathcal {C}}^{-\frac{1}{2}-\kappa }$. For any $F \in \mathrm {Cyl}$, let $\partial _{l_i} F$ denote the Gâteaux derivative of F in direction $l_i$. Let $\nabla F: {\mathcal {C}}^{-\frac{1}{2} -\kappa } \rightarrow {\mathbb {R}}$ be the unique function such that $\partial _{l_i} F(\phi ) = \int _{{\mathbb {T}}_N}\nabla F(\phi ) l_i dx$ for every $\phi \in {\mathcal {C}}^{-\frac{1}{2} - \kappa }$. In other words, $\nabla F$ is the representation of the Gâteaux derivative with respect to the $L^2$ inner product. Then, for any $F,G \in \mathrm {Cyl}$,

$$\begin{aligned} {\mathcal {E}}_{\beta ,N}(F,G) = \Big \langle \int _{{\mathbb {T}}_N}\nabla F \nabla G dx \Big \rangle _{\beta ,N}. \end{aligned}$$

Now we choose a test function in $\mathrm{Cyl}$ to insert into (7.1). Take any $\zeta \in (0,1)$ and $m \in [0, (1-\zeta ){\sqrt{\beta }})$. Let $\chi _m:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a smooth, non-decreasing odd function such that $\chi _m(a) = -1$ for $a \leqslant -m$ and $\chi _m(a) = 1$ for $a \geqslant m$. Define

$$\begin{aligned} F(\phi ) = \chi _m({\mathfrak {m}}_N(\phi )). \end{aligned}$$

Then, $F \in \mathrm {Cyl}$ and $\langle F \rangle _{\beta ,N}= 0$. Moreover, its Fréchet derivative DF is supported on the set $\{ {\mathfrak {m}}_N \in [-m,m] \}$.

Thus, inserting F into (7.1), we obtain

$$\begin{aligned} \lambda _{\beta ,N} \leqslant \frac{{\mathcal {E}}_{\beta ,N}(F,F)}{\langle F^2 \rangle _{\beta ,N}} \leqslant \frac{\Big \Vert \int _{{\mathbb {T}}_N}|\nabla F|^2 dx \Big \Vert _{L^\infty (\nu _{\beta ,N})}}{\langle F^2 \rangle _{\beta ,N}} {\nu _{\beta ,N}}({\mathfrak {m}}_N \in [-m,m]). \end{aligned}$$

(7.2)

For any $g \in L^2({\mathbb {T}}_N)$ and $\varepsilon > 0$, by the linearity of ${\mathfrak {m}}_N$ and the Cauchy–Schwarz inequality,

$$\begin{aligned} \frac{F(\phi +\varepsilon g)-F(\phi )}{\varepsilon }&\leqslant |\chi '_m|_\infty \Big |\frac{{\mathfrak {m}}_N(\phi + \varepsilon g) - {\mathfrak {m}}_N(\phi )}{\varepsilon } \Big | \\&\leqslant |\chi '_m|_\infty \frac{\int _{{\mathbb {T}}_N}g dx}{N^3} \leqslant |\chi '_m|_\infty \frac{ \Big (\int _{{\mathbb {T}}_N}g^2 dx\Big )^\frac{1}{2}}{N^\frac{3}{2}} \end{aligned}$$

where $\chi _m'$ is the derivative of $\chi _m$ and $|\cdot |_\infty$ denotes the supremum norm. Note that this estimate is uniform over $\phi \in {\mathcal {C}}^{-\frac{1}{2} -}$. Then, by duality and the definition of $\nabla F$,

$$\begin{aligned} \begin{aligned} \Bigg \Vert \int _{{\mathbb {T}}_N}|\nabla F|^2 dx \Bigg \Vert _{L^\infty ({\nu _{\beta ,N}})}&= \Bigg \Vert \Big ( \sup _{g \in L^2: \int _{{\mathbb {T}}_N}g^2 dx = 1} \int _{{\mathbb {T}}_N}\nabla F g dx \Big )^2 \Bigg \Vert _{L^\infty ({\nu _{\beta ,N}})} \\&\leqslant \frac{|\chi '_m|_\infty ^2}{N^3}. \end{aligned} \end{aligned}$$

(7.3)

For the other term in (7.2), using that $F^2$ is identically 1 on $\{ |{\mathfrak {m}}_N| \geqslant m \}$,

(7.4)

We insert (7.3) and (7.4) into (7.2) to give

$$\begin{aligned} \lambda _{\beta ,N} \leqslant \frac{|\chi _m|_\infty ^2}{N^3} \frac{\nu _{\beta ,N}({\mathfrak {m}}_N \in [-m,m])}{1 - \nu _{\beta ,N}( {\mathfrak {m}}_N \in (-m,m))}. \end{aligned}$$

By Theorem 1.2, there exists $C=C(\zeta ,\eta )>0$ and $\beta _0 = \beta _0(\zeta ,\eta ) > 0$ such that, for all $\beta > \beta _0$,

$$\begin{aligned} \lambda _{\beta ,N} \leqslant \frac{|\chi _m'|_\infty ^2}{N^3} \frac{e^{-C{\sqrt{\beta }}N^2}}{1-e^{-C{\sqrt{\beta }}N^2}} \end{aligned}$$

from which (1.4) follows. $\quad \square$

Appendix A Analytic Notation and Toolbox

1.1 A. 1 Basic function spaces on the torus

Let ${\mathbb {T}}_N= ({\mathbb {R}}/ N{\mathbb {Z}})^3$ be the 3D torus of sidelength $N \in {\mathbb {N}}$. Denote by $C^\infty ({\mathbb {T}}_N)$ the space of smooth functions on ${\mathbb {T}}_N$ and by $S'({\mathbb {T}}_N)$ the space of distributions. For $\phi \in S'({\mathbb {T}}_N)$ and $f \in C^\infty ({\mathbb {T}}_N)$, we write $\int _{{\mathbb {T}}_N}\phi f dx$ to denote their duality pairing. For any $p \in [1,\infty ]$, let denote the Lebesgue space with respect to the normalised Lebesgue measure .

Let ${\mathcal {F}}$ denote the Fourier transform, i.e. for any $f \in C^\infty ({\mathbb {T}}_N)$ and $n \in (N^{-1}{\mathbb {Z}})^3$,

$$\begin{aligned} {\mathcal {F}}f(n) = \int _{{\mathbb {T}}_N}f e_{-n} d x, \quad f = \frac{1}{N^3} \sum _{n \in (N^{-1}{\mathbb {Z}})^3} {\mathcal {F}}f(n) e_n \end{aligned}$$

where $e_n(x) = e^{2\pi i n \cdot x }$.

For any $\rho : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$, let $T_\rho$ be the Fourier multiplier with symbol $\rho (\cdot )$ defined on smooth functions via

$$\begin{aligned} T_\rho f = \frac{1}{N^3} \sum _{n \in (N^{-1}{\mathbb {Z}})^3} \rho (n) {\mathcal {F}}f(n) e_n. \end{aligned}$$

When clear from context, we simply write $\rho f$ instead of $T_\rho f$.

For $s \in {\mathbb {R}}$, the inhomogeneous Sobolev space $H^s$ is the completion of $f \in C^\infty$ with respect to the norm

$$\begin{aligned} \Vert f \Vert _{H^s} = \Vert \langle \cdot \rangle ^s f \Vert _{L^2} \end{aligned}$$

where $\langle \cdot \rangle = \sqrt{ \eta + 4\pi ^2 | \cdot |^2}$ for a fixed $\eta > 0$ (see Sect. 2). The norms depend on $\eta$ but they are equivalent for different choices.

1.2 A.2 Besov Spaces

In this section, we introduce Besov spaces on ${\mathbb {T}}_N$ and give some useful estimates. All of the results can be found in [BCD11, Section 2.7] stated for Besov spaces on ${\mathbb {R}}^3$, but can be adapted to ${\mathbb {T}}_N$.

Let B(x, r) denote the ball centred at $x \in {\mathbb {R}}^3$ of radius $r > 0$ and let A denote the annulus $B(0, \frac{4}{3}) {\setminus } B(0, \frac{3}{8})$. Let ${\tilde{\Delta }}, \Delta \in C^\infty _c({\mathbb {R}}^3;[0,1])$ be radially symmetric and satisfy

• $\mathrm {supp} {\tilde{\chi }} \subset B(0, \frac{4}{3})$ and $\mathrm {supp} \chi \subset A$;

• $\sum _{k \geqslant -1} \chi _k = 1$, where $\chi _{-1} = {\tilde{\chi }}$ and $\chi _k(\cdot ) = \chi (2^{-k}\cdot )$ for $k \in {\mathbb {N}}\cup \{0\}$.

Identify $\Delta _k$ with its Fourier multiplier.

$\{ \Delta _k \}_{k \in {\mathbb {N}}\cup \{-1\}}$ are called Littlewood-Paley projectors. For $f \in C^\infty ({\mathbb {T}}_N)$, we have

$$\begin{aligned} f = \sum _{k \geqslant -1} \Delta _k f. \end{aligned}$$

For $k \geqslant 0$, $\Delta _k f$ contains the frequencies of f order $2^k$. $\Delta _{-1}$ contains all the low frequencies (i.e. of size less than order 1).

For $s \in {\mathbb {R}}$, $p,q \in [1,\infty ]$, we define the Besov spaces $B^s_{p,q}({\mathbb {T}}_N)$ to be the completion of $C^\infty ({\mathbb {T}}_N)$ with respect to the norm

$$\begin{aligned} \Vert f \Vert _{B^s_{p,q}} = \Big \Vert \Big ( 2^{ks} \Vert \Delta _k f \Vert _{L^p} \Big )_{k \geqslant -1} \Big \Vert _{l^q} \end{aligned}$$

where $l^q$ is the usual space of q-summable sequences, interpreted as a supremum when $q=\infty$. Note that these spaces are separable. Besov-Hölder spaces are denoted $B^s_{\infty ,\infty }({\mathbb {T}}_N) = {\mathcal {C}}^{s}({\mathbb {T}}_N)$ and are a strict subset of the usual Hölder spaces (which are not separable) for $s \in {\mathbb {R}}_+ {\setminus } {\mathbb {N}}$. Moreover, the $B^s_{2,2}({\mathbb {T}}_N) = H^s({\mathbb {T}}_N)$ and their norms are equivalent.

Proposition A.1

(Duality). Let $s \in {\mathbb {R}}$ and $p_1,p_2,q_1,q_2 \in [1,\infty ]$ such that $\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{q_1} + \frac{1}{q_2} = 1$. Then,

(A.1)

for $f,g \in C^\infty ({\mathbb {T}}_N)$.

Proof

See [GOTW18, Lemma 2.1]. $\quad \square$

Proposition A.2

(Fractional Leibniz estimate). Let $s \in {\mathbb {R}}$, $p,p_1,p_2,p_3,p_4,q \in [1,\infty ]$ satisfy $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{p_3} + \frac{1}{p_4}$. Then, there exists $C=C(s,p_1,p_2,p_3,p_4,q,\eta ) > 0$ such that

$$\begin{aligned} \Vert fg \Vert _{B^s_{p,q}} \leqslant C\Vert f \Vert _{B^s_{p_1,q}} \Vert g \Vert _{L^{p_2}} + \Vert f \Vert _{L^{p_3}} \Vert g \Vert _{B^s_{p_4,q}} \end{aligned}$$

(A.2)

for $f,g \in C^\infty ({\mathbb {T}}_N)$.

Proof

See [GOTW18, Lemma 2.1]. $\quad \square$

Proposition A.3

(Interpolation). Let $s,s_1,s_2 \in {\mathbb {R}}$ such that $s_1< s < s_2$, $p,p_1,p_2,q,q_1,q_2 \in [1,\infty ]$ and $\theta \in (0,1)$ satisfy

$$\begin{aligned} s&= \theta s_1 + (1-\theta )s_2 \nonumber \\ \frac{1}{p}&= \frac{\theta }{p_1} + \frac{1-\theta }{p_2} \nonumber \\ \frac{1}{q}&= \frac{\theta }{q_1} + \frac{1-\theta }{q_2}. \end{aligned}$$

(A.3)

Then, there exists $C=C(s,s_1,s_2,p,p_1,p_2,q,q_1,q_2,\theta ,\eta ) > 0$ such that

$$\begin{aligned} \Vert f \Vert _{B^s_{p,q}} \leqslant C\Vert f \Vert _{B^{s_1}_{p_1,q_1}}^\theta \Vert f \Vert _{B^{s_2}_{p_2,q_2}}^{1-\theta } \end{aligned}$$

(A.4)

for $f \in C^\infty ({\mathbb {T}}_N)$.

Proof

See [BM18, Proposition 5.7]. $\quad \square$

Proposition A.4

(Bernstein’s inequality). For $R > 0$, denote $B_f(R) = \{ n \in (N^{-1}{\mathbb {Z}})^3 : |n| \leqslant R \}$. Let $s_1, s_2 \in {\mathbb {R}}$ such that $s_1 < s_2$, $p,q \in [1,\infty ]$. Then, there exists $C=C(s_2,s_2,p,q,\eta )>0$ such that

$$\begin{aligned} \Vert f \Vert _{B^{s_2}_{p,q}}&\leqslant C R^{s_2 - s_1} \Vert f \Vert _{B^{s_1}_{p,q}} \end{aligned}$$

(A.5)

$$\begin{aligned} \Vert g \Vert _{B^{s_1}_{p,q}}&\leqslant CR^{s_1 - s_2}\Vert g \Vert _{B^{s_2}_{p,q}} \end{aligned}$$

(A.6)

for $f,g \in C^\infty ({\mathbb {T}}_N)$ such that $\mathrm {supp} ({\mathcal {F}}f) \subset B_f(R)$ and $\mathrm {supp} ({\mathcal {F}}g) \subset (N^{-1}{\mathbb {Z}})^3 {\setminus } B_f(R)$.

Proof

See [BCD11, Lemma 2.1] for a proof on ${\mathbb {R}}^3$. $\quad \square$

1.3 A.3 Paracontrolled calculus

Let $f,g \in C^\infty ({\mathbb {T}}_N)$. Define the paraproduct

and the resonant product

Then,

(A.7)

Proposition A.5

(Paraproduct estimates). Let $s \in {\mathbb {R}}$ and $p,p_1,p_2,q \in [1,\infty ]$ be such that $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$. Then, there exists $C=C(s,p,p_1,p_2,q,\eta )>0$ such that

(A.8)

for $f,g \in C^\infty ({\mathbb {T}}_N)$.

Proof

See [BCD11, Theorem 2.82] for a proof on ${\mathbb {R}}^3$. $\quad \square$

Proposition A.6

(Resonant product estimate). Let $s_1,s_2 \in {\mathbb {R}}$ such that $s=s_1 + s_2 > 0$. Let $p,p_1,p_2,q \in [1,\infty ]$ satisfy $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$. Then, there exists $C=C(s_1,s_2,p,p_1,p_2,q,\eta ) > 0$ such that

(A.9)

for $f,g \in C^\infty ({\mathbb {T}}_N)$.

Proof

See [BCD11, Theorem 2.85] for a proof on ${\mathbb {R}}^3$. $\quad \square$

We now state some useful commutator estimates.

Proposition A.7

Let $s_1, s_3 \in {\mathbb {R}}$, $s_2 \in (0,1)$ such that $s_1 + s_3 < 0$ and $s_1 + s_2 + s_3 = 0$. Moreover, let $p,p_1,p_2,q_1,q_2 \in [1,\infty ]$ satisfy $\frac{1}{p} + \frac{1}{p_1} + \frac{1}{p_2} = 1$ and $\frac{1}{q_1} + \frac{1}{q_2} = 1$. Then, there exists $C=C(s_1,s_2,s_3,p,p_1,p_2,q_1,q_2,\eta )>0$ such that

(A.10)

for $f,g,h \in C^\infty ({\mathbb {T}}_N)$.

Proof

This is a modification of [GUZ20, Lemma A.6]. See [BG19, Proposition 7]. $\quad \square$

Proposition A.8

Let $s_1,s_3 \in {\mathbb {R}}$, $s_2 \in (0,1)$ such that $s_1+s_3 < 0$ but $s_1+s_2+s_3 > 0$. Morover, let $p,p_1,p_2,p_3 \in [1,\infty ]$ satisfy $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2} + \frac{1}{p_3}$. Then, there exists $C=C(s_1,s_2,s_3,p,p_1,p_2,\eta )>0$ such that

(A.11)

for $f,g,h \in C^\infty ({\mathbb {T}}_N)$.

Proof

This is a modification of [GIP15, Lemma 2.4]. See [BG19, Proposition 6]. $\quad \square$

1.4 A.4 Analytic Properties of ${\mathcal {J}}_k$

The family of operators $\{ {\mathcal {J}}_k \}_{k \geqslant 0}$ defined in Sect. 4.1 satisfies the following estimate: for every multi-index $\alpha \in {\mathbb {N}}^3$, there exists $C=C(\alpha ,\eta )$ such that

$$\begin{aligned} \Big | \partial ^\alpha {\mathcal {J}}_k (x)| \leqslant \frac{C}{\langle k \rangle ^\frac{1}{2} (1+|x|)^{1+|\alpha |}}. \end{aligned}$$

(A.12)

Proposition A.9

Let $s \in {\mathbb {R}}$, $p,q\in [1,\infty ]$. Then, there exists $C=C(s,p,q)>0$ such that

$$\begin{aligned} \Vert {\mathcal {J}}_k f \Vert _{B^{s+1}_{p,q}} \leqslant \frac{C}{\langle k \rangle ^\frac{1}{2}}\Vert f \Vert _{B^s_{p,q}} \end{aligned}$$

(A.13)

for every $f \in C^\infty ({\mathbb {T}}_N)$

Proof

This follows from (A.12) and [BCD11, Proposition 2.78]. $\quad \square$

We now state another useful commutator estimate.

Proposition A.10

Let $s_1 \in {\mathbb {R}}$, $s_2 \in (0,1)$, $p,p_1,p_2,q,q_1,q_2 \in [1,\infty ]$ such that $\frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}$ and $\frac{1}{q} = \frac{1}{q_1} + \frac{1}{q_2}$. Then, for any $\kappa > 0$, there exists $C=C(s_1,s_2,p,p_1,p_2, q, \kappa , \eta )>0$ such that

(A.14)

for $f,g \in C^\infty ({\mathbb {T}}_N)$.

Proof

This follows from (A.12) and [BCD11, Lemma 2.99]. $\quad \square$

1.5 A.5 Poincaré Inequality on Blocks

Proposition A.11

There exists $C_P > 0$ such that, for any $N \in {\mathbb {N}}$ and a unit block, the following estimate holds for all $f \in C^\infty ({\mathbb {T}}_N)$:

(A.15)

where .

Proof

See [GT15, (7.45)]. $\quad \square$

1.6 A.6 Bounds on Discrete Convolutions

Lemma A.12

Let $d \geqslant 1$ and $\alpha , \beta \in {\mathbb {R}}$ satisfy

$$\begin{aligned} \alpha + \beta > d \text { and } \alpha , \beta < d. \end{aligned}$$

Then, there exists $C=C(d,\alpha ,\beta )>0$ such that, uniformly over $n \in (N^{-1}{\mathbb {Z}})^d$,

$$\begin{aligned} \frac{1}{N^3} \sum _{\begin{array}{c} n_1,n_2 \in (N^{-1}{\mathbb {Z}})^d \\ n_1 + n_2 = n \end{array}} \frac{1}{\langle n_1 \rangle ^\alpha \langle n_2 \rangle ^\beta } \lesssim \frac{1}{\langle n \rangle ^{\alpha + \beta - d}} \end{aligned}$$

Proof

Follows from [MWX17, Lemma 4.1] and by keeping track of N dependence.

$\square$