Universality for the 2D Random Walk Loop Soup

TL;DR

This paper proves that the scaling limit of the 2D random walk loop soup converges to the Brownian loop soup on suitable planar graphs, under mild conditions, using Wilson's algorithm and Schramm's theorem.

Contribution

It establishes the universality of the Brownian loop soup as the scaling limit for the 2D random walk loop soup under broad conditions.

Findings

Convergence of the random walk loop soup to the Brownian loop soup is proven.

The proof uses Wilson's algorithm and Schramm's finiteness theorem.

The result applies to graphs with bounded density and crossing estimates.

Abstract

We show that the scaling limit of the random walk loop soup on suitable planar graphs is the Brownian loop soup, under a topology on multisets of unrooted, unparameterized, and macroscopic loops. The result holds assuming only convergence of simple random walk to Brownian motion, a Russo-Seymour-Welsh type crossing estimate, and the bounded density of the graphs. The proof relies on Wilson's algorithm and Schramm's finiteness theorem. Precisely, we approximate the random walk loop soup by the set of loops erased in a greedy variant of Wilson's algorithm, thereby establishing convergence. The resulting limit is identified using the result of Lawler and Ferreras arXiv:math/0409291.