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

Ryoichiro Noda

TL;DR

The paper proves that discrete-time Markov chains on sequences of electrical networks converge, together with their local times, under local Gromov--Hausdorff--vague convergence of the networks plus non-explosion and metric-entropy conditions. The authors develop a discrete-time trace method within an extended Dirichlet-space framework for resistance forms, enabling compact-trace approximations to handle non-compact limits. The results apply to a broad class of models, including critical Galton--Watson trees, uniform spanning trees, random recursive fractals, the critical Erdős--Rényi graph, the configuration model, and random conductance models on fractals, with both deterministic and random network settings. This provides a unified, rigorous approach to scaling limits of discrete processes and their local times on complex networks, with potential implications for algorithmic analyses and probabilistic geometry of random graphs.

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.

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

TL;DR

The paper proves that discrete-time Markov chains on sequences of electrical networks converge, together with their local times, under local Gromov--Hausdorff--vague convergence of the networks plus non-explosion and metric-entropy conditions. The authors develop a discrete-time trace method within an extended Dirichlet-space framework for resistance forms, enabling compact-trace approximations to handle non-compact limits. The results apply to a broad class of models, including critical Galton--Watson trees, uniform spanning trees, random recursive fractals, the critical Erdős--Rényi graph, the configuration model, and random conductance models on fractals, with both deterministic and random network settings. This provides a unified, rigorous approach to scaling limits of discrete processes and their local times on complex networks, with potential implications for algorithmic analyses and probabilistic geometry of random graphs.

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.
Paper Structure (18 sections, 79 theorems, 249 equations, 1 figure)

This paper contains 18 sections, 79 theorems, 249 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The topologies for the main results
  3. The local Gromov--Hausdorff-vague topology
  4. The space $\mathbb{M}_{L}$
  5. Resistance forms
  6. Preliminaries
  7. Extended Dirichlet spaces of resistance forms
  8. Traces of resistance forms
  9. Electrical networks
  10. Resistance forms of electrical networks
  11. Traces of electrical networks
  12. Measurability
  13. Proof of Theorem \ref{['1. thm: deterministic, main result']}
  14. Proof of Theorem \ref{['1. thm: random, main result']}
  15. Applications
  16. ...and 3 more sections

Key Result

Theorem 1.7

Under Assumption 1. assum: deterministic version, the limiting space $G$ belongs to $\check{\mathbb{F}}$ and $\hat{\mathcal{Y}}_{n}$ converges to $\mathcal{X}_{G}$ in $\mathbb{M}_{L}$, where we recall $\mathcal{X}_{G}$ from 1. eq: dfn of cX_G.

Figures (1)

  • Figure 1: From Left to right, the unbounded Sierpiński gasket graphs $G_{0}$, $G_{1}$ and the associated map $g_{m}$.

Theorems & Definitions (182)

  • Definition 1.1: The space $\mathbb{F}$ and $\mathbb{F}_{c}$
  • Definition 1.2: $\varepsilon$-covering, metric entropy
  • Definition 1.3: The space $\check{\mathbb{F}}$ and $\check{\mathbb{F}}_{c}$
  • Definition 1.4: Electrical network
  • Definition 1.5: Resistance forms associated with electrical networks
  • Theorem 1.7
  • Theorem 1.9
  • Remark 1.10
  • Remark 1.11
  • Remark 1.12
  • ...and 172 more