# Talagrand's inequality in planar Gaussian field percolation

## TL;DR

This paper extends percolation results for planar Gaussian fields, proving phase transition, exponential decay, and uniqueness of the infinite cluster using Talagrand's inequality and superconcentration techniques.

## Contribution

It generalizes percolation phase transition results for Gaussian fields by establishing a Talagrand inequality-based superconcentration formula.

## Key findings

- Percolation occurs with probability one for positive levels and not for non-positive levels.
- Exponential decay of crossing probabilities is established.
- Uniqueness of the unbounded cluster is proven.

## Abstract

Let f be a stationary isotropic non-degenerate Gaussian field on R^2. Assume that f = q * W where q is both C^2 and L^2 and W is the L^2 white noise on R^2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that q * q is pointwise non-negative and has fast enough decay, the set {f > -l} percolates with probability one when l > 0 and with probability zero if l < 0 or l = 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 T(g) which is the minimal value such that {g > -T(g)} contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.