Number of distinct sites visited by a resetting random walker

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)> similar to n (d/2) for d < 2 and as < V-0(n)> similar to n for d > 2. For p > 0, we show that < V (p) (n)> grows extremely slowly as similar to[log(n)](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 -> 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.

Automated summary

In this study, we investigate the number of distinct sites visited by resetting random walkers on a d-dimensional hypercubic lattice. We find that the recurrence-transience transition observed in standard random walks disappears with resetting, and we compute the crossover scaling function between different regimes in the limit of small resetting probability. Additionally, we derive the full distribution of visited sites in one dimension and introduce a new observable called imbalance to measure symmetry. Our theoretical results are verified through extensive numerical simulations.