Complete visitation statistics of one-dimensional random walks

Universal exploration dynamics of random walks

“…These multiple-time correlations are crucial to fully characterize the stochastic process { N ( t )}, the number of sites visited at every single time. However, they have been studied only for the special case of 1 d nearest-neighbor RWs 27 , 53 . Using our formalism we can further compute temporal correlations of { N ( t )} for compact Lévy flights in 1 d with 1/ μ = α > 1 (which do leave holes in their trajectories).…”

“…This observable quantifies the efficiency of various stochastic exploration processes, such as animal foraging 24 or the trapping of diffusing molecules 1 , 23 . While the average and, for some examples, the distribution of the number of distinct sites visited, have been determined analytically 5 , 25 – 27 , this information is far from a complete description. In this work, we show that the waiting time τ n , defined as the elapsed time between the visit to the n th and the ( n +1) st distinct, or new, sites characterizes the exploration dynamics in a more fundamental and comprehensive way (Fig.…”

“…When it comes to the exploration of a territory by an animal or biological tracers, a fundamental property in quantifying this exploration is the number N (𝑡) of sites visited by time 𝑡 [1,2]. This quantity has been the focus of many mathematical studies, in which the average, variance, and sometimes the full distribution have been explicitly obtained for a wide range of stochastic processes [3,4,5,6,7,8,9]. These processes include nearest-neighbor jumps on a 𝑑−dimensional lattice, on fractal media, or RWs with long-range jumps, which model a large variety of biological L. Régnier Sorbonne Université, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France e-mail: leo.regnier.pro@outlook.fr M. Dolgushev Sorbonne Université, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France e-mail: maxim.dolgushev@sorbonne-universite.fr O. Bénichou Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), 4 Place Jussieu, 75005 Paris, France e-mail: benichou@lptmc.jussieu.fr motions.…”

Universal exploration dynamics of random walks

“…This observable quantifies the efficiency of various stochastic exploration processes, such as animal foraging [17] or the trapping of diffusing molecules [16,19]. While the average and, for some examples, the distribution of the number of distinct sites visited, have been determined analytically [27,[30][31][32], this information is far from a complete description. In this work, we show that the waiting time τ n , defined as the elapsed time between the visit to the n th and the (n + 1) st distinct, or new, sites characterizes the exploration dynamics in a more fundamental and comprehensive way (Fig.…”

“…Our theoretical approach for the set of inter-visit times τ also represents a start towards determining multipletime visitation correlations for general RWs, quantities that have remained inaccessible thus far. These multipletime correlations have been studied for the very special case of nearest-neighbor RWs, which have no holes in their trajectories [32,53]. Using our formalism we can further compute temporal correlations of {N (t)} for compact Lévy flights in 1d with α > 1 (which do leave holes in their trajectories).…”