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

Abstract

We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on $\Z^4$) 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 $\cmss P_ω^{2n}(0,0)$ 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 $λ_n\to\infty$, such that $$ \cmss P_ω^{2n}(0,0)\ge C(ω)\log(n)n^{-2}/λ_n, $$ with $C(ω)>0$ a.s., along a deterministic subsequence of $n$'s. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the $d\ge5$ 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.

28 pages, version to appear in J. Lond. Math. Soc