[Paper Review] Universality of trap models in the ergodic time scale

M. Jara, Cláudio Landim+1 more|arXiv (Cornell University)|Aug 28, 2012

Stochastic processes and statistical mechanics17 references4 citations

TL;DR

This paper establishes the universality of trap models on a broad class of random graphs—such as the hypercube, d-dimensional torus, random d-regular graphs, and the giant component of super-critical Erdős-Rényi graphs—showing that in the ergodic time scale, the continuous-time random walk with heavy-tailed holding times converges to a K-process under general conditions on the graph structure and i.i.d. vertex weights in the domain of attraction of a stable law with index $\alpha \in (0,1)$. The convergence is proven via scaling limits under mild hypotheses on local geometry and mixing behavior.

ABSTRACT

Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $α$-stable law, $0

Motivation & Objective

• To establish the universality of trap models in the ergodic time scale across diverse random graph ensembles.
• To identify minimal structural and probabilistic conditions on graphs and vertex weights that ensure convergence to the K-process.
• To extend previous results on aging and metastability in trap models to a broad class of graphs, including the hypercube, torus, random regular graphs, and the giant component of super-critical Erdős-Rényi graphs.
• To unify the scaling limit behavior of continuous-time random walks with heavy-tailed holding times across different graph models via a general framework.

Proposed method

• Define a continuous-time random walk on a weighted graph $G_N = (V_N, E_N, W^N)$, where vertex weights $W^N_x$ are i.i.d. and in the domain of attraction of an $\alpha$-stable law with $0 < \alpha < 1$.
• Use a time-rescaling technique to analyze the process in the ergodic time scale, where the system explores the deepest traps.
• Introduce a decomposition of the trajectory into excursions between balls of radius $\ell_N$ around the deepest traps, assuming disjointness and mixing within each ball.
• Apply coupling techniques with Galton-Watson trees to compare local graph structure around typical vertices to infinite trees, ensuring local isomorphism with high probability.
• Verify four key conditions: (B0) deep traps dominate the stationary measure; (B1) local volumes are small; (B2) mixing occurs before hitting the center; (B3) weak transitivity or local isomorphism to trees holds asymptotically.
• Use the K-process as the scaling limit, a continuous-time Markov process on $\mathbb{N}$ that hits any finite set with uniform distribution, capturing aging behavior.

Experimental results

Research questions

• RQ1 Under what general conditions on the graph sequence $G_N$ and vertex weight distribution does the trap model converge to the K-process in the ergodic time scale?
• RQ2 Can the convergence to the K-process be established for graphs beyond the complete graph and $\mathbb{Z}$, such as the hypercube and $d$-dimensional torus?
• RQ3 How do heavy-tailed vertex weights in the domain of attraction of an $\alpha$-stable law with $\alpha < 1$ influence the metastable behavior of the random walk?
• RQ4 What structural properties of $G_N$—such as local geometry, mixing time, and component structure—ensure that the deepest traps govern the long-time behavior?
• RQ5 To what extent can coupling with Galton-Watson trees be used to verify the necessary conditions for convergence in complex random graph models like the giant component of super-critical Erdős-Rényi graphs?

Key findings

• The trap model on $G_N$ converges in law to the $K$-process under the ergodic time scale, provided the vertex weights are i.i.d. and in the domain of attraction of an $\alpha$-stable law with $0 < \alpha < 1$.
• The convergence holds for a broad class of graphs, including the hypercube, $d$-dimensional torus ($d \geq 2$), random $d$-regular graphs, and the largest component of super-critical Erdős-Rényi random graphs.
• The deep traps—vertices with the largest weights—dominate the stationary measure, and their number grows slowly, allowing the system to explore deeper traps over time.
• The key technical condition (B3) is verified for the giant component of super-critical Erdős-Rényi graphs by coupling with Galton-Watson trees and showing that local neighborhoods are isomorphic to infinite trees with high probability.
• The probability of failure in the coupling (i.e., $\mathbb{P}[\mathbb{A}_N^c]$) vanishes as $N \to \infty$, ensuring the validity of the local structure assumptions asymptotically.
• The result establishes universality: despite structural differences, all these graphs yield the same scaling limit—the $K$-process—under the same heavy-tailed weight regime.