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

We study models of discrete-time, symmetric, ZdZd-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances ωxy∈[0,1]ωxy∈[0,1], with polynomial tail near 0 with exponent γ>0γ>0. We first prove for all d≥5d≥5 that the return probability shows an anomalous decay (non-Gaussian) that approaches (up to sub-polynomial terms) a random constant times n−2n−2 when we push the power γγ 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/2n−d/2 for large values of the parameter γγ.