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

TL;DR

This paper proves the convergence of discrete-time Markov chains and their local times on electrical networks under certain topological and metric conditions, with applications to various random graph models.

Contribution

It establishes a general convergence framework for Markov chains on converging electrical networks, extending previous results to complex random structures.

Findings

Convergence of Markov chains on electrical networks under local Gromov--Hausdorff-vague topology.

Application to critical random graph models and fractals.

Characterization of extended Dirichlet spaces for resistance forms.

Abstract

We establish that if a sequence of electrical networks equipped with conductance 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 conditioned on size, uniform spanning trees, random recursive fractals, the critical Erd\H{o}s--R\'{e}nyi 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.