Biased random walks on random graphs

TL;DR

This paper reviews recent mathematical advances in biased random walks on various random graph structures, focusing on their transience, reversibility, and behavior in different environments like $ ext{Z}$, trees, and higher-dimensional lattices.

Contribution

It provides a comprehensive overview of recent developments in the theory of biased random walks on diverse random graphs, emphasizing their properties and behaviors.

Findings

Analysis of transience and recurrence conditions

Characterization of reversible random walks

Insights into behavior on different graph structures

Abstract

These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our main focus will be on directionally transient and reversible random walks on different types of underlying graph structures, such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.