Random walk loop soup

2004/09/16 by Gregory F. Lawler, Lawler, Gregory F., José A. Trujillo Ferreras +1 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.math/0409291

arxiv created 2004/09/16 · openalex publication_date 2004/09/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Brownian loop soup introduced in Lawler and Werner (2004) is a Poissonian realization from a sigma-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlós, Major, and Tusnády. To make the paper self-contained, we include a proof of the approximation result that we need.

Cited by

• Uniform spanning trees and random matrix statistics
• Intensity doubling for Brownian loop-soups in high dimensions
• The real layering field of Brownian loop soup and the Gaussian multiplicative chaos
• On the gap between cluster dimensions of loop soups on ℝ³ and the metric graph of ℤ³
• Arm events in critical planar loop soups

Related

• Universality for the 2D Random Walk Loop Soup
• A Study in Markov Chains, Loop-Erased Random Walk and Loop Soups
• Limit theorems for loop soup random variables
• Conformal Loop Ensembles: Construction via Loop-soups
• Brownian Loops and Conformal Fields