Percolation of discrete GFF in dimension two I. Arm events in the random walk loop soup

TL;DR

This paper investigates the probability of large loop components in a two-dimensional random walk loop soup coming close, using connections to Brownian loop soup and conformal loop ensembles, developing new tools for analysis.

Contribution

It introduces new methods like locality and quasi-multiplicativity for analyzing loop soups, and estimates arm event probabilities in the critical and subcritical regimes.

Findings

Estimated four-arm event probabilities using Brownian loop soup exponents

Developed tools for separation and surgery in loop soup analysis

Established foundational results for connectivity in level sets in a follow-up study

Abstract

In this work, which is the first part of a series of two papers, we study the random walk loop soup in dimension two. More specifically, we estimate the probability that two large connected components of loops come close to each other, in the subcritical and critical regimes. The associated four-arm event can be estimated in terms of exponents computed in the Brownian loop soup, relying on the connection between this continuous process and conformal loop ensembles (with parameter $\kappa \in (8/3,4]$). Along the way, we need to develop several useful tools for the loop soup, based on separation for random walks and surgery for loops, such as a "locality" property and quasi-multiplicativity. The results established here then play a key role in a second paper, in particular to study the connectivity properties of level sets in the random walk loop soup and in the discrete Gaussian free field.