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Random walk with random resetting to the maximum position

Satya N. Majumdar, Sanjib Sabhapandit, and Grégory Schehr
Phys. Rev. E 92, 052126 – Published 19 November 2015

Abstract

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 (1r), it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability (1r)/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<r1, 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 r0 case and we calculate the exact crossover functions that interpolate between the critical (r=0) and the off-critical (r0) behavior for finite but large n.

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  • Received 15 September 2015

DOI:https://doi.org/10.1103/PhysRevE.92.052126

©2015 American Physical Society

Authors & Affiliations

Satya N. Majumdar1, Sanjib Sabhapandit2, and Grégory Schehr1

  • 1LPTMS, CNRS, Univ Paris Sud, Université Paris-Saclay, 91405 Orsay, France
  • 2Raman Research Institute, Bangalore 560080, India

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Issue

Vol. 92, Iss. 5 — November 2015

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