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 ${\mathbb{Z}}^4$) 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 $C(\omega )n^{-2}\log n$, 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 $$ \cmss P_ω^{2n}(0,0)\ge C(ω)\log(n)n^{-2}/λ_n, $$ 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 $d\ge 5$ 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.

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

References in corpus (3)

• Random walks on supercritical percolation clusters
• Recent progress on the Random Conductance Model
• On symmetric random walks with random conductances on ${\mathbb{Z}}^d$

Cited by in corpus (8)

• Recent progress on the Random Conductance Model
• Harnack inequalities on weighted graphs and some applications to the random conductance model
• Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters
• Green kernel asymptotics for two-dimensional random walks under random conductances
• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• Quenched Invariance Principle for a class of random conductance models with long-range jumps
• Trapping in the random conductance model
• Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients