Lower Gaussian heat kernel bounds for the random conductance model in a degenerate ergodic environment

First passage percolation with long-range correlations and applications to random Schrödinger operators

“…Hence, it suffices to consider the special case of the CSRW, for which we have d ω θ = ρ ω by the definition (3.5) so that the bounds in Theorem 3.2 turn immediately into Gaussian estimates with respect to ρ ω . The result follows now, for instance, by the same arguments as in the proof of [10,].…”

“…In this subsection our focus will be on Gaussian heat kernel estimates, see e.g. [28,13,14,35,7,8,10] and references therein for previous results. We recall that, due to a trapping phenomenon, Gaussian bounds do not hold in general: for example, under i.i.d.…”

Upper bounds on the fluctuations for a class of degenerate convex $\nabla ϕ$-interface models

“…These bounds (or the ingredient developed to prove it) became one of the ingredients in the proof of the quenched invariance principle for the random walk on the percolation cluster by Sidoravicius, Sznitman [77], Berger, Biskup [19], Mathieu, Piatnitski [62], the parabolic Harnack inequality and the local limit theorem by Barlow, Hambly [16]. The question of the existence of heat kernel upper and lower bounds (matching the ones of the lattice) have been established for more general degenerate environments satisfying suitable moments assumption by Andres, Deuschel, Slowik [6,7,8] and Andres, Halberstam [9], but this phenomenon is not generic and anomalous heat decay has been proved for some random degenerate environments by Berger, Biskup, Hoffman and Kozma [20], Boukhadra [30], Biskup, Boukhadra [22] and Buckley [36]. Besides the question of the behavior of the heat kernel, the invariance principle has been established for degenerate conductances by Biskup, Prescott [25], Andres, Barlow, Deuschel, Hambly [3], Mathieu [61], Procaccia, Rosenthal, Sapozhnikov [74] and Bella, Schäffner [18].…”

Off-Diagonal Heat Kernel Estimates for Symmetric Diffusions in a Degenerate Ergodic Environment

“…To prove the lower estimate we adapt the established chaining argument to the diffusion in a degenerate random environment, the method originated in [25] using the ideas of Nash. It was adapted to the weighted graph setting in [23], to random walks on percolation clusters in [10], and it was recently applied to the RCM [6]. The strategy is to repeatedly apply lower near-diagonal estimates, derived from the parabolic Harnack inequality established in [19], along a sequence of balls.…”

“…Something stronger than the classical ergodic theorem is required to do this, so given Assumption 1.5 we establish a specific form of concentration inequality (Proposition 3.3) for this purpose. By an argument similar to [6] this inequality is then used to control the environment-dependent terms arising from the Harnack inequality, see Proposition 3.4. The statement is given below and proven in Section 3.…”