A cavity approach to the heterogeneity of the random solid state

Replica statistical mechanics has been invoked to explore a wide collection of properties of the random solid state of matter, as formed, e.g., by vulcanized macromolecular systems. Even at the mean-field level, this approach yields a rich and illuminating description of the state in terms of a fraction of localized particles and the statistical distribution of their localization lengths. An application of the cavity method -- successfully used to analyze a wide range of spin glass models and other statistical-mechanical systems with quenched disorder -- allows replica-based results for random solids to be recovered in a way that sheds light both on their physical origin and their limitations. An extension of this cavity approach, involving an assembly of Ornstein-Uhlenbeck processes, points towards a strategy for addressing certain aspects of the dynamics of the random solid state, at least at the mean-field level.