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Advances in Applied Probability· 2026Q2

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

Ryoichiro Noda

Short summary

Discrete-time Markov chains and their local times on electrical networks converge if the networks converge in a specific topology and meet non-explosion/metric-entropy criteria.

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Abstract

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 Erdős–Rényi 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.

The authors' abstract, as published at the source. Advances in Applied Probability, 2026 · DOI ↗

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Field: Computational Theory and Mathematics

Computational Theory and MathematicsComputer Science