Convergence of the two-dimensional random walk loop soup clusters to CLE

TL;DR

This paper proves that the outer boundaries of clusters in a 2D random walk loop soup converge to the conformal loop ensemble (CLE) in the scaling limit, establishing a connection between discrete models and continuum conformal invariance.

Contribution

It demonstrates the convergence of the discrete loop soup clusters to CLE, extending the understanding of conformal invariance from Brownian models to random walk loop soups.

Findings

Outer boundaries of clusters converge to CLE(kappa)

Relation between kappa and c matches continuum Brownian case

Validates discrete-to-continuum conformal invariance transition

Abstract

We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the CLE(kappa) loop ensemble, with the same relation between kappa and c as in the continuum Brownian setting.