Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances model

TL;DR

This paper investigates the decay of return probabilities for random walks in random environments with conductances following a power law near zero, establishing the standard spectral dimension as correct for certain parameters.

Contribution

It demonstrates that for conductance exponents greater than half the dimension, the standard spectral dimension accurately describes the decay of return probabilities.

Findings

Standard bound is of correct logarithmic order for b3 > d/2

Results apply to both continuous-time and discrete-time models

Spectral dimension remains valid in the specified regime

Abstract

We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $\gamma$ near 0. We are interested in estimating the quenched decay of the return probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. We show that for $\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order. As an expected concequence, the same result holds for the discrete-time case.