An Analysis of the Recurrence/Transience of Random Walks on Growing Trees and Hypercubes

TL;DR

This paper investigates how the growth rate of infinite graphs influences whether a simple random walk on them is recurrent or transient, revealing a phase transition dependent on the graph's growth speed.

Contribution

It introduces a specific model of random walk on growing graphs and establishes a phase transition between recurrence and transience based on growth speed, using a novel coupling argument.

Findings

Phase transition between recurrence and transience depending on growth speed.

Development of a coupling method called less homesick as graph growing (LHaGG).

Examples include random walks on growing hypercubes and binary sequences.

Abstract

It is a celebrated fact that a simple random walk on an infinite $k$-ary tree for $k \geq 2$ returns to the initial vertex at most finitely many times during infinitely many transitions; it is called transient. This work points out the fact that a simple random walk on an infinitely growing $k$-ary tree can return to the initial vertex infinitely many times, it is called recurrent, depending on the growing speed of the tree. Precisely, this paper is concerned with a simple specific model of a random walk on a growing graph (RWoGG), and shows a phase transition between the recurrence and transience of the random walk regarding the growing speed of the graph. To prove the phase transition, we develop a coupling argument, introducing the notion of less homesick as graph growing (LHaGG). We also show some other examples, including a random walk on $\{0,1\}^n$ with infinitely growing $n$, of the phase transition between the recurrence and transience. We remark that some graphs concerned in this paper have infinitely growing degrees.