Chemical distances for percolation of planar Gaussian free fields and critical random walk loop soups

TL;DR

This paper investigates the chemical distances within percolation clusters of level sets of 2D Gaussian free fields and loop clusters from random walk loop soups, revealing critical properties and new proof techniques.

Contribution

It introduces the study of chemical distances in these models and establishes that at criticality, the distance scales linearly with positive probability, using novel proof methods.

Findings

Chemical distance between macroscopic annuli is of dimension 1 at criticality.

Proof combines Makarov's theorem, isomorphism theory, and entropic repulsion estimates.

Provides new insights into percolation cluster geometry in Gaussian free fields and loop soups.

Abstract

We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a hard wall.