Title:	Random walk with random resetting to the maximum position.
Authors:	Majumdar, Satya N. Sabhapandit, Sanjib Schehr, Gregory
Issue Date:	Nov-2015
Publisher:	American Physical Society
Citation:	Physical Review E, 2015, Vol 92, p052126
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 (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.
Description:	Open access
URI:	http://hdl.handle.net/2289/6397
ISSN:	2470-0053 (online) 2470-0045
Alternative Location:	http://arxiv.org/abs/1509.04516 http://dx.doi.org/10.1103/PhysRevE.92.052126
Copyright:	2015 American Physical Society
Appears in Collections:	Research Papers (TP)

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