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

TL;DR

This paper investigates the decay rate of the heat kernel for a four-dimensional random walk with i.i.d. conductances, revealing a logarithmic correction to the known polynomial decay and demonstrating the influence of trapping phenomena across multiple scales.

Contribution

It constructs environments showing the logarithmic factor in heat-kernel decay is genuine, extending understanding of trapping effects in four-dimensional random conductance models.

Findings

The heat kernel decay includes a logarithmic correction term.

Construction of environments with arbitrarily slow decay along subsequences.

The results hold alongside a quenched invariance principle.

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 $\cmss P_\omega^{2n}(0,0)$ after $2n$ steps is at most $C(\omega) n^{-2} \log n$, but the best lower bound till now has been $C(\omega) n^{-2}$. Here we will show that the $\log n$ term marks a real phenomenon by constructing an environment, for each sequence $\lambda_n\to\infty$, such that $$ \cmss P_\omega^{2n}(0,0)\ge C(\omega)\log(n)n^{-2}/\lambda_n, $$ with $C(\omega)>0$ 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.