Dynamics of unvisited sites in presence of mutually repulsive random walkers

TL;DR

This paper studies the persistence probability of unvisited sites in a lattice with mutually repulsive random walkers, revealing a stretched exponential decay and mapping to independent walkers under certain conditions.

Contribution

It introduces a mapping between interacting and independent walkers' persistence problems, providing new insights into their dynamics and decay behavior.

Findings

Persistence probability decays as a stretched exponential with exponent ~0.5.

Mapping to independent walkers is possible with a specific density transformation.

Derived analytical form for the persistence in the interacting walkers' system.

Abstract

We have considered the persistence of unvisited sites of a lattice, i.e., the probability $S(t)$ that a site remains unvisited till time $t$ in presence of mutually repulsive random walkers. The dynamics of this system has direct correspondence to that of the domain walls in a certain system of Ising spins where the number of domain walls become fixed following a zero termperature quench. Here we get the result that $S(t) \propto \exp(-\alpha t^{\beta})$ where $\beta$ is close to 0.5 and $\alpha$ a function of the density of the walkers $\rho$. The number of persistent sites in presence of independent walkers of density $\rho^\prime$ is known to be $S^\prime (t) = \exp(-2 \sqrt{\frac{2}{\pi}} \rho^\prime t^{1/2})$. We show that a mapping of the interacting walkers' problem to the independent walkers' problem is possible with $\rho^\prime = \rho/(1-\rho)$ provided $\rho^\prime, \rho$ are small. We also discuss some other intricate results obtained in the interacting walkers' case.