Number of distinct sites visited by a resetting random walker

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 $⟨V_0(n)⟩\sim n^{d/2}$ for $d<2$ and as $⟨V_0(n)⟩\sim n$ for $d>2$. For $p>0$, we show that $⟨V_p(n)⟩$ 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.

41 pages, 8 figs

References in corpus (21)

• First Passage Under Restart
• First order transition for the optimal search time of Lévy flights with resetting
• Diffusion in a potential landscape with stochastic resetting
• Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results
• Diffusion with resetting in arbitrary spatial dimension
• Dynamical transition in the temporal relaxation of stochastic processes under resetting
• Random walks with preferential relocations to places visited in the past and their application to biology
• Stochastic Search with Poisson and Deterministic Resetting
• Monotonous continuous-time random walks with drift and stochastic reset events
• Localization transition induced by learning in random searches
• Stochastic resetting in underdamped Brownian motion
• Extremal statistics for stochastic resetting systems
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Record statistics for random walks and Lévy flights with resetting
• Random acceleration process under stochastic resetting
• Diffusive transport on networks with stochastic resetting to multiple nodes
• Distribution of the time of the maximum for stationary processes
• Spatial Extent of Branching Brownian Motion
• Comparison of two models of tethered motion
• Exact distributions of cover times for $N$ independent random walkers in one dimension
• Resetting in Stochastic Optimal Control

Cited by in corpus (12)

• Discrete space-time resetting model: Application to first-passage and transmission statistics
• Random walks on complex networks under time-dependent stochastic resetting
• Universal exploration dynamics of random walks
• Striking universalities in stochastic resetting processes
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Entropy rate of random walks on complex networks under stochastic resetting
• Non-homogeneous random walks with stochastic resetting: an application to the Gillis model
• From Maximum of Intervisit Times to Starving Random Walks
• Diffusion with stochastic resetting on a lattice
• The distribution of the maximum of independent resetting Brownian motions
• Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials
• Evanescent random walker on networks: Hitting times, budget renewal, and survival dynamics