Recent progress on the Random Conductance Model

TL;DR

This paper reviews recent advances in the Random Conductance Model, focusing on scaling limits, effective resistance behavior, and gradient field models with non-convex interactions, highlighting key theoretical developments in the field.

Contribution

It provides a comprehensive overview of recent results on the scaling limits and behavior of the Random Conductance Model, including new insights into non-convex gradient fields.

Findings

Scaling limit results for random walks in random conductance environments

Observations on effective resistance behavior

Analysis of non-convex gradient field models

Abstract

Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.