Aging and sub-aging for Bouchaud trap models on resistance metric spaces

TL;DR

This paper demonstrates that Bouchaud trap models on converging electrical networks exhibit aging and sub-aging behaviors, with results applicable to various low-dimensional graph structures.

Contribution

It establishes convergence and aging properties of BTMs on resistance metric spaces under local Gromov-Hausdorff convergence, extending to diverse graph classes.

Findings

BTMs converge and age on networks with Gromov-Hausdorff convergence.

Sub-aging occurs when local structures of networks converge.

Results apply to Sierpiński gasket, Galton-Watson trees, Erdős-Rényi graphs.

Abstract

In this paper, we prove that if a sequence of electrical networks converges in the local Gromov-Hausdorff topology and satisfies a non-explosion condition, then the associated Bouchaud trap models (BTMs) also converge and exhibit aging. Moreover, when local structures of electrical networks converge, we prove sub-aging. Our results are applicable to a wide class of low-dimensional graphs, including the two-dimensional Sierpi\'{n}ski gasket, critical Galton-Watson trees, and the critical Erd\H{o}s-R\'{e}nyi random graph. The proof consists of two main steps: Polish metrization of the vague-and-point-process topology and showing the precompactness of transition densities of BTMs.