2015/09/15 by Satya N. Majumdar, Sanjib Sabhapandit, Grégory Schehr +1 · 3 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Computer science #Current (fluid) #Diffusion and Search Dynamics #Jumping #Mathematics #Path (computing) #Physics #Position (finance) #Random walk #Reset (finance) #Statistical physics #Statistics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.92.052126
published as Phys. Rev. E 92, 052126 (2015) · 14 pages, 7 figures
arxiv created 2015/09/15 · openalex publication_date 2015/11/19 · arxiv updated 2015/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability r, and with probability (1-r), it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability (1-r)/2. For r=0, it reduces to a standard random walk whose typical distance grows as √n for large n. In the presence of a nonzero resetting rate 0<r≤1, we find that both the average maximum and the average position grow ballistically for large n, with a common speed v(r). Moreover, the fluctuations around their respective averages grow diffusively, again with the same diffusion coefficient D(r). We compute v(r) and D(r) explicitly. We also show that the probability distribution of the difference between the maximum and the location of the walker becomes stationary as n→∞. However, the approach to this stationary distribution is accompanied by a dynamical phase transition, characterized by a weakly singular large deviation function. We also show that r=0 is a special "critical" point, for which the growth laws are different from the r→0 case and we calculate the exact crossover functions that interpolate between the critical (r=0) and the off-critical (r→0) behavior for finite but large n.