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

We study models of discrete-time, symmetric, -valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances [omega]xy[set membership, variant][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].