Scaling limits of discrete-time Markov chains and their local times on electrical networks

In this paper, we establish that if a sequence of electrical networks equipped with natural measures converges in the local Gromov-Hausdorff-vague topology and satisfies certain non-explosion and metric-entropy conditions, then the sequence of associated discrete-time Markov chains and their local times also converges. This result applies to many examples, such as critical Galton-Watson trees, uniform spanning trees, random recursive fractals, the critical Erd\"os-R\'enyi random graph, the configuration model, and the random conductance model on fractals. To obtain the convergence result, we characterize and study extended Dirichlet spaces associated with resistance forms, and we study traces of electrical networks.