Heat-kernel estimates for random walk among random conductances with heavy tail

Details Heat-kernel estimates for random walk among random conductances with heavy tail Heat-kernel estimates for random walk among random conductances with heavy tail

2008-12-14

arxiv.org/abs/0812.2669v4

Stochastic Processes and their Applications 120 (2010) 182-194

Mathematics

Probability

Version to appear in SPA

Scientific paper

We study models of discrete-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$, with polynomial tail near 0 with exponent $\gamma>0$. We first prove for all $d\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $\gamma$.

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