A Study in Markov Chains, Loop-Erased Random Walk and Loop Soups

TL;DR

This paper explores the mathematical relationships between Markov chains, loop-erased random walks, and loop soups, introducing new models and connections in stochastic processes and graph theory.

Contribution

It defines new concepts like transient and recurrent chains with killing, and introduces the growing loop model, linking LERW with spanning trees and loop measures.

Findings

Established connections between LERW and UST

Defined new models for loop measures and soups

Explored properties of Markov chains with killing

Abstract

In this paper, we make a few random explorations that relate directly to the items mentioned in the title. We define transient chains and recurrent chains with "killing", the Green's function, the Laplacian operator, and harmonic functions. We then introduce the loop-erased random walk (LERW) and its relationship with the uniform spanning tree (UST). We finish by introducing loop measures and soups and defining the "growing loop" model.