Random walk loop soup

Abstract:

The Brownian loop soup introduced by 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.