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Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg–Landau ∇ ϕ Interface Model

Published version
Peer-reviewed

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Type

Article

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Authors

Taylor, Peter A. 

Abstract

Abstract: We study a continuous-time random walk on Zd in an environment of random conductances taking values in (0, ∞). 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 ∇ϕ 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.

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Keywords

Article, Random conductance model, Local limit theorem, De Giorgi iteration, Ergodic, Stochastic interface model, 60K37, 60J35, 60F17, 82C41, 39A12

Journal Title

Journal of Statistical Physics

Conference Name

Journal ISSN

0022-4715
1572-9613

Volume Title

182

Publisher

Springer US