Number of distinct sites visited by a resetting random walker

Abstract

We investigate the number V p (n) of distinct sites visited by an nstep 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 asWe 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 → 0, we compute the exact crossover scaling function between the two regimes. In the onedimensional 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.

Random walks on complex networks under node-dependent stochastic resetting

“…We have derived the exact expressions of stationary occupation probabilities of the walker on each node and the MFPT between arbitrary two nodes. The two quantities (see equations (11) and ( 17)) can be calculated from the matrix Z defined in equation (12). Our deduction is general and is able to apply any protocol of node-dependent resetting probability.…”

“…The mean perimeter and the mean area of the convex hull of a two-dimensional resetting Brownian motion were exactly computed, which showed the two quantities grow much more slowly with time than the case without resetting [10]. For random walks on a d-dimensional hypercubic lattice under resetting [11], the average number of distinct sites visited by the walker grows extremely slowly with the time steps, and the so-called recurrencetransience transition at d = 2 for standard random walks (without resetting) disappears in the presence of resetting. In a finite one-dimensional domain, the distribution of the number of distinct sites visited by a random walker before hitting a target site with and without resetting was deduced, and the distribution can be simply expressed in terms of splitting probabilities only [12].…”

Complete Visitation Statistics of 1d Random Walks

“…It bears emphasizing that P(N (t)) is a single-time quantity-the distribution of the number of distinct sites visited at one time instant. Although this distribution is known for the nearest-neighbor random walk in one dimension [5][6][7][8][9][10], its continuous counterpart [11,12] and more involved processes (see for instance [13][14][15][16]), it provides limited information about the full stochastic process {N (t)}, where the braces denote the set of N (t) values for each time step of the walk (Fig. 1).…”