Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nabla \varphi$ Interface Model

Abstract

We study a continuous-time random walk on ${\mathbb{Z}}^d$ in an environment of random conductances taking values in $(0,\infty )$. For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem is then applied to prove a scaling limit result for the space-time covariances in the Ginzburg-Landau $\nabla \varphi$ model. We also show that the associated Gibbs distribution scales to a Gaussian free field. These results apply to convex potentials for which the second derivative may be unbounded.

37 pages, accepted version, to appear in J. Stat. Phys

References in corpus (4)

• Quantitative stochastic homogenization and large-scale regularity
• Anomalous heat-kernel decay for random walk among bounded random conductances
• Limit theory for random walks in degenerate time-dependent random environments
• Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights

Cited by in corpus (8)

• Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights
• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• Quenched Invariance Principle for a class of random conductance models with long-range jumps
• The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Random partitions under the Plancherel-Hurwitz measure, high genus Hurwitz numbers and maps
• The Discrete Gaussian model, I. Renormalisation group flow at high temperature
• A Ray-Knight theorem for $\nabla \varphi$ interface models and scaling limits