Abstract
We consider the random walk on a simple point process on ℝd, d≥2, whose jump rates decay exponentially in the α-power of jump length. The case α=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 α∈(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 L2. For the Poisson point process, we prove that at α=d, there is a transition from diffusive to subdiffusive behavior of the mixing time.
Citation
Pietro Caputo. Alessandra Faggionato. "Isoperimetric inequalities and mixing time for a random walk on a random point process." Ann. Appl. Probab. 17 (5-6) 1707 - 1744, October 2007. https://doi.org/10.1214/07-AAP442
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