Number of distinct sites visited by a resetting random walker

Abstract

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)⟩ ∼ n d/2 for d < 2 and as ⟨V 0(n)⟩ ∼ n for d > 2. For p > 0, we show that ⟨V p (n)⟩ grows extremely slowly as ∼ 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.