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 $S(t)\propto \exp (-\alpha t^{\beta })$ 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.

6 pages, 7 figures

References in corpus (1)

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