We consider the random walk on a simple point process on R-d, d >= 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 a epsilon (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 a = d, there is a transition from diffusive to subdiffusive behavior of the mixing time.
Isoperimetric inequalities and mixing time for a random walk on a random point process / Pietro, Caputo; Faggionato, Alessandra. - In: THE ANNALS OF APPLIED PROBABILITY. - ISSN 1050-5164. - STAMPA. - 17:5-6(2007), pp. 1707-1744. [10.1214/07-aap442]
Isoperimetric inequalities and mixing time for a random walk on a random point process
FAGGIONATO, ALESSANDRA
2007
Abstract
We consider the random walk on a simple point process on R-d, d >= 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 a epsilon (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 a = d, there is a transition from diffusive to subdiffusive behavior of the mixing time.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.