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

We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $\omega_{xy}>0$ 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 $P_\omega^{2n}(0,0)$. We prove that $P_\omega^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.Comment: 22 pages. Includes a self-contained proof of isoperimetric inequality for supercritical percolation clusters. Version to appear in AIHP + additional correction

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on $\Z^4$) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability \cmss P_\omega^{2n}(0,0) after $2n$ steps is at most $C(\omega) n^{-2} \log n$, but the best lower bound till now has been $C(\omega) n^{-2}$. Here we will show that the $\log n$ term marks a real phenomenon by constructing an environment, for each sequence $\lambda_n\to\infty$, such that \cmss P_\omega^{2n}(0,0)\ge C(\omega)\log(n)n^{-2}/\lambda_n, with $C(\omega)>0$ a.s., along a deterministic subsequence of $n$'s. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the $d\ge5$ cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.Comment: 28 pages, version to appear in J. Lond. Math. So

Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments

We introduce anchored versions of the Nash inequality. They allow to control the $L^2$ norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments. As an example, we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium.Comment: 18 page

Trapping in the random conductance model

We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.Comment: 21 pages, version to appear in J. Statist. Phy

Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results by the authors to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.Comment: 19 pages; accepted version, to appear in Electron. Commun. Proba

Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance Model

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.Comment: To appear in Journal of Math. Soc. Japan, special volum