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

ABSTRACT. We consider the nearest-neighbor simple random walk on Zd, 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 Zd. 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 P2n ω (0, 0). We prove that P2n ω (0, 0) is bounded by a random constant times n−d/2 in d = 2, 3, while it is o(n−2) in d ≥ 5 and O(n−2 log n) in d = 4. By producing examples with anomalous heat-kernel decay approaching 1/n2 we prove that the o(n−2) 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. 1