Spectra of random stochastic matrices and relaxation in complex systems

Abstract

We compute spectra of large stochastic matrices W, defined on sparse random graphs in the configuration model class, i.e. on graphs that are maximally random subject to a given degree distribution. Edges of the graph are given positive random weights Wij > 0 in such a fashion that column sums are normalized to one. We compute spectra of such matrices both in the thermodynamic limit, and for single large instances. Our results apply to arbitrary graphs in the configuration model class, as long as the mean vertex degree remains finite in the thermodynamic limit. Edge weights Wij can be largely arbitrary but we require the Wij to satisfy a detailed balance condition, or in other words that the Markov chains described by them are reversible. Knowing the spectra of stochastic matrices is tantamount to knowing the complete spectrum of relaxation times of stochastic processes described by them, so our results should have many interesting applications for the description of relaxation in complex systems. Contributions to the spectral density related to extended states can be disentangled from those related localized states allowing time scales associated with transport processes and those associated with the dynamics of local rearrangements to be differentiated.