Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $$\mathbb Z^d$$ Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.

Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

We establish a quenched local central limit theorem for the dynamic random conductance model on $${\mathbb {Z}}^d$$ Z d only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi’s iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate ergodic weights.

Phase transitions for a class of gradient fields

We consider gradient fields on $${\mathbb {Z}}^d$$ Z d for potentials V that can be expressed as $$\begin{aligned} e^{-V(x)}=pe^{-\frac{qx^2}{2}}+(1-p)e^{-\frac{x^2}{2}}. \end{aligned}$$ e - V ( x ) = p e - q x 2 2 + ( 1 - p ) e - x 2 2 . This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. We then show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential V. Based on these results we can give a new proof for the non-uniqueness of ergodic zero-tilt gradient Gibbs measures in dimension 2. In contrast to the first proof of this result we rely on planar duality and do not use reflection positivity. Moreover, we show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of p and q and, in dimension $$d\ge 4$$ d ≥ 4 , for q close to one or for $$p(1-p)$$ p ( 1 - p ) sufficiently small.