Number of distinct sites visited by a resetting random walker – Archive ouverte HAL

Marco Biroli ¹ Francesco Mori ¹ Satya N. Majumdar ¹ Satya Majumdar ¹

Marco Biroli, Francesco Mori, Satya N. Majumdar, Satya Majumdar. Number of distinct sites visited by a resetting random walker. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2022, 55 (24), pp.244001. ⟨10.1088/1751-8121/ac6b69⟩. ⟨hal-03721604⟩

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.

• 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques