Aging dynamics and the topology of inhomogenous networks

TL;DR

This paper investigates how the aging dynamics in phase ordering processes are influenced by network topology, revealing that the aging exponent relates to the network's critical properties and spectral dimension.

Contribution

It establishes a general relation between the aging exponent and network topology, including critical and lower critical dimension cases, supported by numerical results on various networks.

Findings

The aging exponent $a_\chi$ is positive above the lower critical dimension.

For networks at the lower critical dimension, $a_\chi$ vanishes.

In $O(N)$ models, $a_\chi$ depends on the spectral dimension.

Abstract

We study phase ordering on networks and we establish a relation between the exponent $a_\chi$ of the aging part of the integrated autoresponse function $\chi_{ag}$ and the topology of the underlying structures. We show that $a_\chi >0$ in full generality on networks which are above the lower critical dimension $d_L$, i.e. where the corresponding statistical model has a phase transition at finite temperature. For discrete symmetry models on finite ramified structures with $T_c = 0$, which are at the lower critical dimension $d_L$, we show that $a_\chi$ is expected to vanish. We provide numerical results for the physically interesting case of the $2-d$ percolation cluster at or above the percolation threshold, i.e. at or above $d_L$, and for other networks, showing that the value of $a_\chi $ changes according to our hypothesis. For $O({\cal N})$ models we find that the same picture holds in the large-${\cal N}$ limit and that $a_\chi$ only depends on the spectral dimension of the network.