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We prove upper bounds on the transition probabilities of random walks withi.i.d. random conductances with a polynomial lower tail near $0$. We considerboth constant and variable speed models. Our estimates are sharp. As aconsequence, we derive local central limit theorems, parabolic Harnackinequalities and Gaussian bounds for the heat kernel. Some of the arguments arerobust and applicable for random walks on general graphs. Such results arestated under a general setting.

Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance Model