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

We consider the nearest-neighbor simple random walk on Zd, d 2, driven by a field of bounded random conductances xy [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of xy > 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 (0, 0). We prove that P2n (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.

• Type: Journal
• Year: 2006
• Where: CORR
• Authors: Noam Berger, Marek Biskup, Christopher E. Hoffman, Gady Kozma
• Keywords: CORR 2006, Education, Random, Random Conductances, Simple Random Walk