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 , and with probability , it undergoes symmetric random walk, i.e., it hops to one of its neighboring sites, with equal probability . For , it reduces to a standard random walk whose typical distance grows as for large . In the presence of a nonzero resetting rate , we find that both the average maximum and the average position grow ballistically for large , with a common speed . Moreover, the fluctuations around their respective averages grow diffusively, again with the same diffusion coefficient . We compute and explicitly. We also show that the probability distribution of the difference between the maximum and the location of the walker becomes stationary as . 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 is a special “critical” point, for which the growth laws are different from the case and we calculate the exact crossover functions that interpolate between the critical and the off-critical behavior for finite but large .
- Received 15 September 2015
DOI:https://doi.org/10.1103/PhysRevE.92.052126
©2015 American Physical Society