Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

Abstract

We investigate level-set percolation of the discrete Gaussian free field on ${\mathbb {Z}}^d$, $d\ge 3$, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level $\alpha$ disconnects the discrete blow-up of a compact set $A\subseteq {\mathbb {R}}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of A, when disconnection occurs. These bounds, combined with the findings of the recent work by Duminil-Copin, Goswami, Rodriguez and Severo, show that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of A. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain ‘profile’ description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of A, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the ‘solidification estimates’ developed in Nitzschner and Sznitman (to appear in J Eur Math Soc, arXiv:1706.07229).

Appendix A. Strong coupling lemma

In this appendix we state and prove Proposition A.1 which provides a uniform comparison between the discrete harmonic potential of an arbitrary finite union of discrete L-boxes and the Brownian potential of their ${\mathbb {R}}^d$-filling when L converges to infinity quick enough. The proof relies on a strong coupling result of [13] in the spirit of Komlós, Major and Tusnády.

We first introduce some notation. We consider integers $L = L(N) \ge 1$, $N\in {\mathbb {N}}$. We define for a non-empty finite set of points ${\mathcal {C}}\subseteq {\mathbb {Z}}^d$ the set

$$\begin{aligned} C = \bigcup _{z\in {\mathcal {C}}} \Big (z + [0,L)^d \Big )\cap {\mathbb {Z}}^d, \end{aligned}$$

(A.1)

and its ${\mathbb {R}}^d$-filling, that is the compact set

$$\begin{aligned} \Gamma = \bigcup _{z\in {\mathcal {C}}} \Big (z + [0,L]^d\Big ). \end{aligned}$$

(A.2)

Moreover we define the compact sets

$$\begin{aligned} {\widetilde{\Gamma }} =\big \{x\in \Gamma \,;\, d_\infty (x,\partial \Gamma )\ge L/4\big \},\quad {\widehat{\Gamma }} = \big \{x\in {\mathbb {R}}^d\,;\, d_\infty (x,\Gamma )\le L/4 \big \}. \end{aligned}$$

(A.3)

Then, we have the following comparison between the hitting probabilities of C for the discrete random walk and the hitting probabilities of ${\widetilde{\Gamma }}$ and ${\widehat{\Gamma }}$ for the Brownian motion.

Proposition A.1

Assume that $c' N^\eta \le L \le c'' N$ for some $\eta >0$. Then, for any fixed integer $M \ge 1$

$$\begin{aligned}&\liminf _{N\rightarrow \infty } \inf _{{\mathcal {C}}\subseteq B(0, M N)} \inf _{x\in {\mathbb {Z}}^d} \Big (P_x[ H_{C}<\infty ] - W_x[H_{{\widetilde{\Gamma }}}<\infty ]\Big ) \ge 0, \end{aligned}$$

(A.4)

$$\begin{aligned}&\limsup _{N\rightarrow \infty } \sup _{{\mathcal {C}}\subseteq B(0, M N)} \sup _{x\in {\mathbb {Z}}^d}\Big (P_x[ H_{C}<\infty ] - W_x[H_{{\widehat{\Gamma }}}<\infty ]\Big ) \le 0. \end{aligned}$$

(A.5)

Proof

The argument for this proposition is largely inspired by the proof of Proposition 4.1 of [19]. We will only show (A.4) as (A.5) can be proven in a similar way interchanging the roles of the simple random walk and Brownian motion.

Fix $R>M+c''$ so that ${\widetilde{\Gamma }} \subseteq B_\infty (0, R N)$ for all ${\mathcal {C}}\subseteq B(0,M N)$. We can write

$$\begin{aligned} \begin{aligned} \inf _{{\mathcal {C}}\subseteq B(0, M N)}\inf _{x\in {\mathbb {Z}}^d}&\Big ( P_x[ H_{C}<\infty ] - W_x[H_{{\widetilde{\Gamma }}}<\infty ]\Big ) \\&\ge \left( \inf _{{\mathcal {C}}\subseteq B(0, M N)}\inf _{x\in B(0,R N)} \Big ( P_x[ H_{C}<\infty ] - W_x[H_{{\widetilde{\Gamma }}}<\infty ]\Big ) \right) \\&\wedge \Big ( - \sup _{z\in B_\infty (0,R)^c} W_z[H_{B(0,M+c'')}<\infty ] \Big ), \end{aligned}\nonumber \\ \end{aligned}$$

(A.6)

where we used scaling invariance of the Brownian motion to obtain the term in the second line. Note that the second member of the minimum does not depend on N and converges to zero as $R \rightarrow \infty$ using the transience of Brownian motion. We will now show that the limit of the first member of the minimum as N grows to infinity is non-negative.

For a closed set $F \subseteq {\mathbb {R}}^d$ we denote by $L_F = \sup \{0<t<\infty ; Z_t\in F\}$ the time of last visit of the Brownian motion to F (using the convention that $L_F = 0$ if the set on the right-hand side is empty). Clearly, for all $x\in B(0, R N)$ and all ${\mathcal {C}}\subseteq B(0,MN)$, we have ${\widetilde{\Gamma }}\subseteq B(x,2NR)$. Thus, for any fixed $\epsilon >0$

$$\begin{aligned} W_x[H_{{\widetilde{\Gamma }}}<\infty ]\le W_x[H_{{\widetilde{\Gamma }}}\le N^{2+\epsilon }] + W_x [L_{B(x,2NR)}> N^{2+\epsilon }], \end{aligned}$$

(A.7)

where the second summand on the right-hand side converges to 0 as $N\rightarrow \infty$ uniformly in $x\in B(0, R N)$ and ${\mathcal {C}}\subseteq B(0,MN)$. In fact, $W_x [L_{B(x,2NR)}> N^{2+\epsilon }] = W_0 [L_{B(0,2R)}> N^{\epsilon }] \rightarrow 0$ as $N\rightarrow \infty$ by scaling invariance and transience of the Brownian motion. Similar to [19, Proposition 4.1], we define ${\widehat{H}}_{{\widetilde{\Gamma }}}$ to be the smallest integer multiple of 1 / d bigger or equal than $H_{{\widetilde{\Gamma }}}$. Applying the strong Markov property at $H_{{\widetilde{\Gamma }}}$ and using translation invariance we get

$$\begin{aligned} W_x[H_{{\widetilde{\Gamma }}}\le N^{2+\epsilon }]\le & {} W_x\Big [H_{{\widetilde{\Gamma }}}\le N^{2+\epsilon }, |Z_{H_{{\widetilde{\Gamma }}}}-Z_{{\widehat{H}}_{{\widetilde{\Gamma }}}}|_{\infty }\le \frac{L}{8}\Big ] \nonumber \\&+\, W_0\Big [\sup _{0\le t\le 1/d}|Z_t|_{\infty }\ge \frac{L}{8}\Big ]. \end{aligned}$$

(A.8)

Note that $W_0[\sup _{0\le t\le 1/d}|Z_t|_{\infty }\ge L/8]\rightarrow 0$ as $N\rightarrow \infty$. By Theorem 4 of [13] and the fact that $L\ge c'N^\eta$ for some $\eta >0$, there exists a probability space $(\overline{\Omega },\overline{{\mathcal {F}}},{\overline{P}})$, a simple random walk $({\overline{X}}_n)_{n\ge 0}$ on ${\mathbb {Z}}^d$ and a Brownian motion $({\overline{Z}}_t)_{t\ge 0}$ on ${\mathbb {R}}^d$, both started at x, such that

$$\begin{aligned} {\overline{P}}\Big [\max _{1\le k\le 2d N^{2+\epsilon }} | {\overline{X}}_k - {\overline{Z}}_{k/d}|_\infty < \frac{L}{8}\Big ] \rightarrow 0, \quad \text {as }N\rightarrow \infty . \end{aligned}$$

(A.9)

Denote by $H_{{\widetilde{\Gamma }}}^{{\overline{Z}}}$ and $H_{C}^{{\overline{X}}}$ the entrance times in ${\widetilde{\Gamma }}$ and C, associated to ${\overline{Z}}_\cdot$ and ${\overline{X}}_\cdot$ respectively. By combining (A.7)–(A.9), we obtain

$$\begin{aligned} \begin{aligned} W_x[H_{{\widetilde{\Gamma }}}< \infty ]&\le {\overline{P}}\Big [H_{{\widetilde{\Gamma }}}\le N^{2+\epsilon }, \max _{1\le k\le 2d N^{2+\epsilon }} | {\overline{X}}_k - {\overline{Z}}_{k/d}|_\infty< \tfrac{L}{8}, \\&\qquad \qquad \qquad |{\overline{Z}}_{H^{{\overline{Z}}}_{{\widetilde{\Gamma }}}}-{\overline{Z}}_{{\widehat{H}}^{{\overline{Z}}}_{{\widetilde{\Gamma }}}}|_{\infty }\le \tfrac{L}{8}\Big ] + o(1)\\&\le {\overline{P}}[H_{C}^{{\overline{X}}} < \infty ] + o(1), \quad \text {as }N\rightarrow \infty , \end{aligned} \end{aligned}$$

(A.10)

uniformly in ${\mathcal {C}}\subseteq B(0,MN)$ and in $x \in B(0,RN)$. The result now follows by plugging (A.10) in (A.6) and by sending first $N\rightarrow \infty$ and then $R\rightarrow \infty$. $\square$