Upper bounds on the one-arm exponent for dependent percolation models

TL;DR

This paper establishes upper bounds on the one-arm exponent for dependent percolation models, extending classical results to models with dependencies like Gaussian fields, using novel exploration and entropy techniques.

Contribution

It introduces a new approach based on exploration and relative entropy to derive bounds for dependent percolation models, including Gaussian fields.

Findings

Proves $ta_1 \u2264 1/3$ for Gaussian fields in 2D with rapid decay

Establishes $ta_1 \u2264 d/3$ for finite-range fields in $d  3$

Demonstrates sharp phase transition and mean-field bounds using a new Russo-type inequality

Abstract

We prove upper bounds on the one-arm exponent $\eta_1$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson-Voronoi and Poisson-Boolean percolation. More precisely, in dimension $d=2$ we prove that $\eta_1 \le 1/3$ for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in $d \ge 3$ we prove $\eta_1 \le d/3$ for finite-range fields, both discrete and continuous, and $\eta_1 \le d-2$ for fields with rapid correlation decay. Although these results are classical for Bernoulli percolation (indeed they are best-known in general), existing proofs do not extend to dependent percolation models, and we develop a new approach based on exploration and relative entropy arguments. The proof also makes use of a new Russo-type inequality for Gaussian fields, which we apply to prove the sharpness of the phase transition and the mean-field bound for finite-range fields.