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.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
9 articles.
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