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