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Convergence of blanket times for sequences of random walks on critical random graphs

Part of: Limit theorems Graph theory Special processes Markov processes

Published online by Cambridge University Press:  09 January 2023

George Andriopoulos
George Andriopoulos*
Affiliation:
NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, China.
*
Email: gandriopoulos91@gmail.com
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Abstract

Under the assumption that sequences of graphs equipped with resistances, associated measures, walks and local times converge in a suitable Gromov-Hausdorff topology, we establish asymptotic bounds on the distribution of the $\varepsilon$ -blanket times of the random walks in the sequence. The precise nature of these bounds ensures convergence of the $\varepsilon$ -blanket times of the random walks if the $\varepsilon$ -blanket time of the limiting diffusion is continuous at $\varepsilon$ with probability 1. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees and the Erdős-Rényi random graph in the critical window. We highlight that proving continuity of the $\varepsilon$ -blanket time of the limiting diffusion relies on the scale invariance of a finite measure that gives rise to realizations of the limiting compact random metric space, and therefore we expect our results to hold for other examples of random graphs with a similar scale invariance property.

Keywords

random walk in random environmentblanket timeGromov-Hausdorff convergenceGalton-Watson treeErdős-Rényi random graph

MSC classification

Primary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles 05C81: Random walks on graphs
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C80: Random graphs 60J25: Continuous-time Markov processes on general state spaces
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 3 , May 2023 , pp. 478 - 515
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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