Anomalous heat-kernel decay for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on Z, d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy > 0 exceeds the threshold for bond percolation on Z. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2n-step return probability P ω (0, 0). We prove that P 2n ω (0, 0) is bounded by a random constant times n in d = 2, 3, while it is o(n) in d ≥ 5 and O(n log n) in d = 4. By producing examples with anomalous heat-kernel decay approaching 1/n we prove that the o(n) bound in d ≥ 5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d = 4.