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

Pietro Caputo, Alessandra Faggionato

2007
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The Annals of Applied Probability
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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
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... ^{-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.

doi:10.1214/07-aap442
fatcat:apts3gp73bfotk56y7qtcuz42y