%0 Journal Article %1 caputo2007isoperimetric %A Caputo, Pietro %A Faggionato, Alessandra %D 2007 %I Institute of Mathematical Statistics %J The Annals of Applied Probability %K Cheeger_constant Poisson_random_environment isoperimetric_inequality long_distance_dispersal mixing_time random_walk_in_random_environment %N 5/6 %P pp. 1707-1744 %T Isoperimetric Inequalities and Mixing Time for a Random Walk on a Random Point Process %U http://www.jstor.org/stable/25442904 %V 17 %X We consider the random walk on a simple point process on $\Bbb R^d$, 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.

@article{caputo2007isoperimetric, abstract = {We consider the random walk on a simple point process on ${\Bbb R}^{d}$, 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.}, added-at = {2013-11-04T22:25:27.000+0100}, author = {Caputo, Pietro and Faggionato, Alessandra}, biburl = {https://www.bibsonomy.org/bibtex/2efd37f1702debde2069498556082fe3f/peter.ralph}, copyright = {Copyright © 2007 Institute of Mathematical Statistics}, interhash = {dd16bcf59e017b67fa6400e5299c78e3}, intrahash = {efd37f1702debde2069498556082fe3f}, issn = {10505164}, journal = {The Annals of Applied Probability}, jstor_articletype = {research-article}, jstor_formatteddate = {Oct. - Dec., 2007}, keywords = {Cheeger_constant Poisson_random_environment isoperimetric_inequality long_distance_dispersal mixing_time random_walk_in_random_environment}, language = {English}, number = {5/6}, pages = {pp. 1707-1744}, publisher = {Institute of Mathematical Statistics}, timestamp = {2013-11-04T22:25:27.000+0100}, title = {Isoperimetric Inequalities and Mixing Time for a Random Walk on a Random Point Process}, url = {http://www.jstor.org/stable/25442904}, volume = 17, year = 2007 }