Advances in Applied Probability· 2026Q2
Scaling limits of discrete-time Markov chains and their local times on electrical networks
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- Q2SCImago
- 2026year
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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Key points
- Convergence of discrete-time Markov chains and local times is proven for converging electrical networks.
- Convergence requires specific non-explosion and metric-entropy conditions on the networks.
- The results generalize to various random graph and fractal models.
- The method involves characterizing extended Dirichlet spaces and network traces.
AI-generated from the title and abstract; the full text is not read.
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