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

Abstract

We prove upper bounds on the one-arm exponent η1 for dependent percolation models; while our main interest is level set percolation of smooth 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 η1 ≤ 1/3 for Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in general dimensions we prove η1 ≤ d/3 for finite-range fields and η1 ≤ 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. We also establish a new Russo-type inequality for smooth Gaussian fields which we use to prove the sharpness of the phase transition for finite-range fields.

Mean-field bounds for Poisson-Boolean percolation

“…In this section we establish the main entropic bounds (see Proposition 2.1) that will underpin the proof of Theorems 1.2-1.4. These are similar to bounds proven in [7] in the setting of Bernoulli and Gaussian percolation.…”

“…In this paper we show how to adapt this method to the Poisson-Boolean model, where certain extra technical difficulties arise. As well as the applications in the current paper, we expect that similar applications to those developed in [7,17] could also be implemented for the Poisson-Boolean model using this approach. We expect this method can also be adapted to other models in the Bernoulli universality class, such as Poisson-Voronoi percolation.…”

“…This strategy was introduced by the authors in a previous work [7], where it was used to give bounds on the one-arm exponent for Bernoulli percolation and other Gaussian dependent percolation models. It was subsequently exploited in [17] where it used to show, in the context of Bernoulli percolation, that mean-field behaviour of the susceptibility implies mean-field behaviour of the infinite cluster density and critical cluster volume.…”

“…The stopping time lemma. The proof of Proposition 2.1 relies on an exact formula for the relative entropy between stopped sequences of independent random variables; this is a generalisation of [7,Lemma 2.11] which considered the i.i.d. case.…”

Variance bounds for Gaussian first passage percolation

“…In particular, Condition 2.5 is then verified. Now for the lower bound, we will make use of the following decomposition proved by S. Muirhead and the author in a previous work [6]. Definition 3.7.…”