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

TL;DR

This paper studies the asymptotic behavior of blanket times for random walks on sequences of critical random graphs, establishing convergence results under certain topological assumptions and highlighting the role of scale invariance.

Contribution

It provides a framework for proving convergence of blanket times on critical random graphs using Gromov-Hausdorff topology and scale invariance properties.

Findings

Established bounds on the distribution of ε-blanket times.

Proved convergence of blanket times for critical Galton-Watson trees, Erdős-Rényi graphs, and configuration models.

Highlighted the importance of scale invariance in the limiting diffusion.

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 with probability one at $\varepsilon$. This result enables us to prove annealed convergence in various examples of critical random graphs, including critical Galton-Watson trees, the Erd\H{o}s-R\'enyi random graph in the critical window and the configuration model in the scaling 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.