Walks in rough energy landscapes: a network model

A simple way to describe the slow relaxation and ageing of a glass is to consider the system as a point in configuration-space hopping between local energy minima. The associated Markov process is specified by the distribution of these minima and the transition rates between them. Previous studies have explored the analytically tractable mean-field case [Bouchaud et al, 96] where the network of allowed transitions is fully connected. We consider a more elaborate version of the model by introducing the concept of distance among minima: the evolution takes place on sparse networks. This brings the problem into the realm of sparse random matrices. We therefore base our analysis on the spectral properties of the infinitesimal generator of the process - the master operator. We use the cavity method to evaluate the average eigenvalue spectrum and degree of localisation of eigenstates in the thermodynamic limit. These quantities are key in determining the dynamics of the system and can be used to compute time-dependent observables such as the return probability. Our findings show that eigenstates have attributes arising from a non-trivial combination of the corresponding mean field and infinite temperature limits of the model, indicating the existence of three different regimes in time.