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

We consider the nearest-neighbor simple random walk on ℤ^d , d ≥ 2, driven by a field of bounded random conductances ω_xy ∞ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ω_xy > 0 exceeds the threshold for bond percolation on ℤ^d . 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_ω²ⁿ (0, 0) .We prove that P_ω²ⁿ (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 andO(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.