Isoperimetric inequalities and mixing time for a random walk on a random point process

Abstract

We consider the random walk on a simple point process on ${\mathbb{R}}^d$, $d\ge 2$, whose jump rates decay exponentially in the $\alpha$-power of jump length. The case $\alpha =1$ corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for $\alpha \in (0,d)$, that the random walk confined to a cubic box of side $L$ has a.s. Cheeger constant of order at least $L^{-1}$ and mixing time of order $L^2$. For the Poisson point process, we prove that at $\alpha =d$, there is a transition from diffusive to subdiffusive behavior of the mixing time.

Published in at http://dx.doi.org/10.1214/07-AAP442 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Determinantal Processes and Independence
• Quenched invariance principle for simple random walk on percolation clusters
• Quenched invariance principles for random walks with random conductances
• Mott law as lower bound for a random walk in a random environment
• Isoperimetric inequalities and mixing time for a random walk on a random point process
• Quenched invariance principles for random walks on percolation clusters

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• Diffusivity in one-dimensional generalized Mott variable-range hopping models
• Invariance principle for Mott variable range hopping and other walks on point processes
• Stochastic homogenization in amorphous media and applications to exclusion processes
• Recurrence and transience for long-range reversible random walks on a random point process
• Growing in time IDLA cluster is recurrent