Aging dynamics and the topology of inhomogenous networks

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 $O(\mathcal{N})$ models we find that the same picture holds in the large-$\mathcal{N}$ limit and that $a_{\chi }$ only depends on the spectral dimension of the network.

arXiv commentLateX file, 4 eps figures

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• Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz-Thouless phase
• Fluctuations and effective temperatures in coarsening
• Phase ordering in disordered and inhomogeneous systems
• Topological regulation of activation barriers on fractal substrates
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• Phase ordering and universality for continuous symmetry models on graphs
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• Non-equilibrium critical properties of the Ising model on product graphs
• Phase ordering in 3d disordered systems
• Complex phase-ordering of the one-dimensional Heisenberg model with conserved order parameter
• Growth Kinetics and Aging Phenomena in a Frustrated System
• Effects of Frustration on Fluctuation-dissipation Relations
• Diffusion on asymmetric fractal networks