Random walk with random resetting to the maximum

TL;DR

This paper analyzes a one-dimensional random walk with a novel resetting rule to the maximum visited site, revealing ballistic growth of maximum and position, explicit speed and diffusion coefficients, and a phase transition in the distribution of their difference.

Contribution

It introduces and analytically solves a new random walk model with maximum-based resetting, deriving growth laws, explicit coefficients, and phase transition phenomena.

Findings

Average maximum and position grow ballistically with rate v(r).

Fluctuations around averages grow diffusively with coefficient D(r).

Stationary distribution of the difference between maximum and position exists, with a phase transition.

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 $\sqrt{n}$ for large $n$. In presence of a nonzero resetting rate $0<r\le 1$, 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 $n\to \infty$. 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 $r\to 0$ case and we calculate the exact crossover functions that interpolate between the critical $(r=0)$ and the off-critical $(r\to 0)$ behavior for finite but large $n$.