Dynamics of unvisited sites in presence of mutually repulsive random walkers

Details Dynamics of unvisited sites in presence of mutually repulsive random walkers Dynamics of unvisited sites in presence of mutually repulsive random walkers

2007-03-06

arxiv.org/abs/cond-mat/0703144v1

Physics

Condensed Matter

Statistical Mechanics

6 pages, 7 figures

Scientific paper

10.1088/1751-8113/40/23/001

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.

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