Percolation for two-dimensional excursion clouds and the discrete Gaussian free field

TL;DR

This paper investigates percolation properties of excursion processes and the discrete Gaussian free field in the plane, establishing critical parameters and their relationships through SLE computations and isomorphism theorems.

Contribution

It proves the equality of critical percolation parameters for excursion clouds and their scaling limits, and establishes bounds and inequalities for the dGFF critical parameters.

Findings

Critical parameters for vacant set percolation are equal to π/3.

The critical level for the dGFF is positive and less than √(π/2).

A strict inequality h*<√(2u*) is established between the critical parameters.

Abstract

We study percolative properties of excursion processes and the discrete Gaussian free field (dGFF) in the planar unit disk. We consider discrete excursion clouds, defined using random walks as a two-dimensional version of random interlacements, as well as its scaling limit, defined using Brownian motion. We prove that the critical parameters associated to vacant set percolation for the two models are the same and equal to $\pi/3.$ The value is obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism theorem, we use a generalization of the discrete result that also involves a loop soup (and an SLE computation) to show that the critical parameter associated to level set percolation for the dGFF is strictly positive and smaller than $\sqrt{\pi/2}.$ In particular this entails a strict inequality of the type $h_*<\sqrt{2u_*}$ between the critical percolation parameters of the dGFF and the two-dimensional excursion cloud. Similar strict inequalities are conjectured to hold in a general transient setup.