Anomalous Heat Kernel for Random Walks in Random Environments of Conductances
Omar Boukhadra
TL;DR
This work analyzes the anomalous heat-kernel decay for random walks in random environments of conductances on $\mathbb{Z}^d$, focusing on i.i.d. conductances with a polynomial tail near zero and, separately, a polynomial tail at infinity. The authors develop a trapping-based framework and combine coarse-graining with spectral (Feynman-Kac) methods to obtain matching lower and upper bounds on the return probability $P^{2n}_{\omega}(o,o)$, expressing it in terms of a trap probability $\Theta_n$. They identify dimension-dependent thresholds, notably $\alpha_c=\frac{1}{4}\frac{d-4}{2d-1}$ for the $[0,1]$ regime and $\alpha_v=\frac{1}{4}\frac{d-4}{d}$ in the $[1,\infty)$ regime, separating anomalous from normal behavior, with $d=4$ exhibiting normal diffusion and the symmetric model displaying opposite trends between the two walkers. The results advance understanding of trapping phenomena in disordered media and provide precise asymptotics linking environmental traps to long-time transport properties.
Abstract
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.