Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances model

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.