Spiders in random environment

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 $ω=(ω_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $κ$ 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 $κ/N>1$ and null if $κ/N<1$. In particular, if $κ/N <1$ a spider has null speed but the speed of a (single) RWRE is positive.

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