How fast does the range of simple random walk grow?
Itai Benjamini, Justin Salez
TL;DR
The paper investigates how the range $R_t$ of a discrete-time simple random walk grows on general infinite graphs. By expressing the growth via discovery times $T_n$ and introducing two coarse geometric parameters $f(n)$ (edge density) and $g(n)$ (volume growth), the authors derive a universal bound $\mathbb{E}[T_n] \le 4 n f(n) \sum_{r=0}^{n-1} \frac{1}{g(r)}$, which implies $\mathbb{E}[T_n] \le 4 n^3 \log n$ and $\mathbb{E}[R_t] \ge c (t/\log t)^{1/3}$. They establish near-optimality via a multi-scale Feige-type Lollipop graph, showing $\mathbb{E}[T_n] \gtrsim n^3$ for dyadic $n$, and discuss oscillations between slow and fast range growth. The work also presents a simple uniform transience condition guaranteeing linear range growth and posits a conjecture that vertex-nonamenable graphs should exhibit linear range, linking isoperimetric properties to the dynamics of range growth. Overall, the paper provides a unified framework for understanding range growth that bridges local geometry and global walk behavior on general graphs.
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 $T_n:=\inf\{t\ge 0: R_t\ge n\}$ 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.