How fast does the range of simple random walk grow?

TL;DR

This paper investigates the growth rate of the range of simple random walks on graphs, establishing bounds, constructing sharp examples, and exploring conditions for linear growth, with implications for understanding random walk behavior on complex networks.

Contribution

It provides universal bounds on the expected discovery time and range growth, constructs a graph with near-sharp bounds, and introduces a transience condition for linear range growth.

Findings

Expected discovery time $ o$ at most $4n^3 ext{log} n$

Range growth $ o$ at least $(t/ ext{log} t)^{1/3}$

Constructed graph with $ o$ $ ext{E}[T_n] ext{ } ext{gtrsim} ext{ } n^3$

Abstract

Consider a discrete-time simple random walk $(X_t)_{t\ge 0}$ on an infinite, connected, locally finite graph $G$. Let $R_t := |\{X_0,\dots,X_t\}|$ denote its range at time $t$, and $T_n:=\inf\{t\ge 0: R_t= n\}$ the $n-$th discovery time. We establish a general estimate on $\mathbb E[T_n]$ in terms of two coarse geometric parameters of $G$, and deduce the universal bounds $\mathbb E[T_n]\le 4n^3\log n$ and $\mathbb E[R_t]\gtrsim (t/\log t)^{1/3}$. Moreover, we show that this is essentially sharp by constructing a multi-scale version of Feige's Lollipop graph satisfying $\mathbb E[T_n]\gtrsim n^{3}$ for all dyadic integers $n$. In light of this example, we ask whether the existence of \emph{trapping phases} where the range grows sub-diffusively necessarily implies the existence of \emph{expanding phases} where it grows super-diffusively. Finally, we provide a simple \emph{uniform transience} condition under which the expected range grows linearly, and conjecture that all vertex-nonamenable graphs exhibit linear range.