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

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) is an element of [0, 1], with polynomial tail near 0 with exponent gamma > 0. We first prove for all d >= 5 that the return probability shows an anomalous decay (non- Gaussian) that approaches (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. (C) 2009 Elsevier B.V. All rights reserved.