Dynamics of unvisited sites in presence of mutually repulsive random walkers

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 $S(t) \propto \exp(-αt^β)$ where $β$ is close to 0.5 and $α$ a function of the density of the walkers $ρ$. The number of persistent sites in presence of independent walkers of density $ρ^\prime$ is known to be $S^\prime (t) = \exp(-2 \sqrt{\frac{2}π} ρ^\prime t^{1/2})$. We show that a mapping of the interacting walkers' problem to the independent walkers' problem is possible with $ρ^\prime = ρ/(1-ρ)$ provided $ρ^\prime, ρ$ are small. We also discuss some other intricate results obtained in the interacting walkers' case.

6 pages, 7 figures