John Thacker

Abstract: The Brownian loop soup introduced by Lawler and Werner is a Poissonian realization from a particular \sigma-finite measure on loops in C.  This measure satisfies conformal invariance and a restriction property.  The random walk loop soup introduced by Lawler and Trujillo Ferreras is a realization from a similar measure on random walk loops in C and converges to the Brownian loop soup.  In this dissertation, we present several results on Brownian loop soup and random walk loop soup.  First, we show that the complement of the Brownian loop soup (i.e., the set of points not surrounded by any loop) has a Hausdorff dimension of 2 – \lambda/5, where \lambda is the intensity of the loop soup.  Next, we extend the definitions of Brownian and random walk loop soup to arbitrary higher dimension and show that the random walk loop soup can also be considered a measure on continuous time random walk loops.  We then give a strong approximation that shows that the random walk loop soup converges to the Brownian loop soup in any dimension, making use of the strong approximation result of Komlós, Major, and Tusnády.