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I will consider the random walk on Z^d driven by a field of random i.i.d. conductances. The law of the conductances is bounded from above; no restriction is posed on the lower tail (at zero) except that the bonds with positive conductances percolate. I will explain how the quenched invariance principle is proved for this random walk despite the fact that the (quenched) heat kernel may exhibit anomalous decay in time. I will also derive universal upper bounds on the heat-kernel decay which, as I will show, can be saturated by appropriately chosen conductance distributions. Based on joint work with N. Berger, C. Hoffman, G. Kozma and T. Prescott