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

We study models of discrete-time, symmetric, ${\mathbb{Z}}^d$-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 $d\ge 5$ 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 $n^{-d\mathrm{/}2}$ for large values of the parameter $\gamma$.