Number of distinct sites visited by a resetting random walker

TL;DR

This paper studies how resetting affects the growth of the number of distinct sites visited by a random walker, revealing a logarithmic growth for resetting cases and analyzing distributions and new observables.

Contribution

It provides exact results for the growth of visited sites under resetting, including crossover functions and distributions, and introduces the imbalance measure for one-dimensional walks.

Findings

Visited sites grow as (log n)^d for p>0

Recurrence-transience transition disappears with resetting

Full distribution of visited sites and imbalance derived

Abstract

We investigate the number $V_p(n)$ of distinct sites visited by an $n$-step resetting random walker on a $d$-dimensional hypercubic lattice with resetting probability $p$. In the case $p=0$, we recover the well-known result that the average number of distinct sites grows for large $n$ as $\langle V_0(n)\rangle\sim n^{d/2}$ for $d<2$ and as $\langle V_0(n)\rangle\sim n$ for $d>2$. For $p>0$, we show that $\langle V_p(n)\rangle$ grows extremely slowly as $\sim \left[\log(n)\right]^d$. We observe that the recurrence-transience transition at $d=2$ for standard random walks (without resetting) disappears in the presence of resetting. In the limit $p\to 0$, we compute the exact crossover scaling function between the two regimes. In the one-dimensional case, we derive analytically the full distribution of $V_p(n)$ in the limit of large $n$. Moreover, for a one-dimensional random walker, we introduce a new observable, which we call imbalance, that measures how much the visited region is symmetric around the starting position. We analytically compute the full distribution of the imbalance both for $p=0$ and for $p>0$. Our theoretical results are verified by extensive numerical simulations.