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 ω x y ∈ [ 0 , 1 ] , with polynomial tail near 0 with exponent γ > 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 γ 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 γ .