Spiders in random environment

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 $\mathbb{Z}$ as underlying random walk. We suppose the environment $\omega =({\omega }_x{)}_{x\in \mathbb{Z}}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $\kappa$ such that $\E[((1-ω_0)/ω_0)^κ]=1$. 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.

25 pages, 5 figures

References in corpus (2)

• Molecular Spiders with Memory
• On slowdown and speedup of transient random walks in random environment

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