Universality for the 2D Random Walk Loop Soup

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 Trujillo Ferreras arXiv:math/0409291.