Number of distinct sites visited by a resetting random walker
Marco Biroli, Francesco Mori, Satya N. Majumdar
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo>∼</mml:mo> <mml:mspace width="-4pt"/> <mml:msup> <mml:mrow> <mml:mfenced close="]" open="["> <mml:mrow> <mml:mi>log</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> . 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.