Talagrand’s inequality in planar Gaussian field percolation

Abstract

Let f be a stationary isotropic non-degenerate Gaussian field on ${\mathbb{R}}^2$. Assume that $f=q\ast W$ where $q\in L^2({\mathbb{R}}^2)\cap C^2({\mathbb{R}}^2)$ and W is the $L^2$ white noise on ${\mathbb{R}}^2$. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that $q\ast q$ is pointwise non-negative and has fast enough decay, the set $\{f\ge -\ell \}$ percolates with probability one when $\ell >0$ and with probability zero if $\ell \le 0$. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcentration formula for the threshold $\text{T}(g)$ which is the minimal value such that $\{g\ge -\text{T}(g)\}$ contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.