Spiders in random environment

TL;DR

This paper investigates the movement speed of a multi-particle 'spider' in a one-dimensional random environment, showing that the spider's speed is positive or zero depending on the ratio of environment parameter to number of particles.

Contribution

It establishes a phase transition in the spider's speed based on the ratio of environment parameter to the number of particles, extending understanding of RWRE dynamics.

Findings

Spider speed is positive if ppa/N > 1.

Spider speed is zero if ppa/N < 1.

Single RWRE has positive speed, unlike the spider when ppa/N < 1.

Abstract

A spider consists of several, say $N$, particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on $\Z$ as underlying random walk. We suppose the environment $\omega=(\omega_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $\kappa$ such that $\E[((1-\omega_0)/\omega_0)^{\kappa}]=1$. The restriction rules are kept very general; we only assume transitivity and irreducibility of the spider. The main result is that the speed of a spider is positive if $\kappa/N>1$ and null if $\kappa/N<1$. In particular, if $\kappa/N <1$ a spider has null speed but the speed of a (single) RWRE is positive.