Sharp phase transition for Gaussian percolation in all dimensions

TL;DR

This paper proves a sharp phase transition in Gaussian percolation models across all dimensions, showing exponential decay below criticality and percolation in slabs above, extending 2D results to higher dimensions with novel techniques.

Contribution

It establishes a universal sharp phase transition for Gaussian percolation in all dimensions using a new comparison method with discretized and truncated models.

Findings

Exponential decay of connection probabilities below critical level

Percolation in thick slabs above critical level

Extension of 2D results to arbitrary dimensions

Abstract

We consider the level-sets of continuous Gaussian fields on $\mathbb{R}^d$ above a certain level $-\ell\in \mathbb{R}$, which defines a percolation model as $\ell$ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than $d$ (in particular, this includes the Bargmann-Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point $\ell_c$. More precisely, we show that connection probabilities decay exponentially for $\ell<\ell_c$ and percolation occurs in sufficiently thick 2D slabs for $\ell>\ell_c$. This extends results recently obtained in dimension $d=2$ to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice $\varepsilon\mathbb{Z}^d$) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter $\ell$.