Random walk loop soup

Random Walk Loop Soup

The Brownian loop soup introduced in [3] is a Poissonian realization from a σ-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, M...

Loop-Erased Random Walk

Scaling Limit of Loop-erased Random Walk

The loop-erased random walk (LERW) was first studied in 1980 by Lawler as an attempt to analyze self-avoiding walk (SAW) which provides a model for the growth of a linear polymer in a good solvent. The self-avoiding walk is simply a path on a lattice that does not visit the same site more than once. Proving things about the collection of all such paths is a formidable challenge to rigorous math...

3 The Brownian loop soup

We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to “chronologically add Brownian loops” to simple curves in the plane.

A Random Walk with Exponential Travel Times

Consider the random walk among N places with N(N - 1)/2 transports. We attach an exponential random variable Xij to each transport between places Pi and Pj and take these random variables mutually independent. If transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...