Compact directed percolation with movable partial reflectors

TL;DR

This paper investigates a one-dimensional compact directed percolation model with movable partial reflectors, revealing how the survival probability decays over time with a continuously varying exponent depending on reflection probability.

Contribution

It introduces a novel model linking compact directed percolation with random walkers and movable reflectors, analyzing survival probability decay with a variable exponent.

Findings

Survival probability decays as a power law with exponent between 1/2 and 1.160.

The decay exponent varies continuously with reflection probability.

The model maps to a pair of unbiased random walkers with movable partial reflectors.

Abstract

We study a version of compact directed percolation (CDP) in one dimension in which occupation of a site for the first time requires that a "mine" or antiparticle be eliminated. This process is analogous to the variant of directed percolation with a long-time memory, proposed by Grassberger, Chate and Rousseau [Phys. Rev. E 55, 2488 (1997)] in order to understand spreading at a critical point involving an infinite number of absorbing configurations. The problem is equivalent to that of a pair of random walkers in the presence of movable partial reflectors. The walkers, which are unbiased, start one lattice spacing apart, and annihilate on their first contact. Each time one of the walkers tries to visit a new site, it is reflected (with probability r) back to its previous position, while the reflector is simultaneously pushed one step away from the walker. Iteration of the discrete-time evolution equation for the probability distribution yields the survival probability S(t). We find that S(t) \sim t^{-delta}, with delta varying continuously between 1/2 and 1.160 as the reflection probability varies between 0 and 1.