Anomalous Heat Kernel for Random Walks in Random Environments of Conductances

We study the trapping phenomenon of random walks in random environments of i.i.d. random conductances on the bonds of the grid $\mathbb{Z}^d$, the so-called random conductance model. Our main results concern the important model with conductances in $[0, 1]$ and a polynomial-tailed law near zero for which we find the correct order of decay of the anomalous heat kernel for $d \ge 5$. In $d = 4$, the behavior is found to be normal. In addition, we look at the symmetrical situation with conductances in $[1, \infty)$ with a polynomial law at infinity, which also shows opposite return probability behaviors.