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

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

2006-07-31

arxiv.org/abs/math/0607805v2

Annals of Applied Probability 2007, Vol. 17, No. 5,6, 1707-1744

Mathematics

Probability

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

Scientific paper

10.1214/07-AAP442

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, 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.

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