Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances model

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We study models of continuous-time, symmetric, ᵈ-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances ωxy∈[0,1] with a power law with an exponent γ near 0. We are interested in estimating the quenched decay of the return probability Pωᵗ(0,0), as t tends to +∞. We show that for γ> 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.