Subdiffusive heat‐kernel decay in four‐dimensional i.i.d. random conductance models

We study the diagonal heat‐kernel decay for the four‐dimensional nearest‐neighbor random walk (on ℤ4) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability Pω2n(0,0) after 2n steps is at most C(ω) n−2 log n, but the best lower bound till now has been C(ω) n−2. Here we will show that the log n term marks a real phenomenon by constructing an environment, for each sequence λn→∞, such that Pω2n(0,0)⩾C(ω)log(n)n−2/λn, with C(ω)>0 almost surely, along a deterministic subsequence of n's. Notably, this holds simultaneously with a (non‐degenerate) quenched invariance principle. As for the d ⩾ 5 cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.