Exact solution of dynamical mean-field theory for a linear system with annealed disorder

Abstract We investigate a disordered multi-dimensional linear system in which the interaction parameters are colored noises, varying stochastically in time with defined temporal correlations. We refer to this type of disorder as ‘annealed’, in contrast to quenched disorder in which couplings are fixed over time. Using generating functional methods, we extend dynamical mean-field theory to accommodate annealed disorder and employ it to find the exact solution of the linear model in the limit of a large number of degrees of freedom. Our analysis yields analytical results for the non-stationary autocorrelation, the stationary variance, the power spectral density, and the phase diagram of the model. Some unexpected features emerge upon changing the correlation time of the interactions. The stationary variance of the system and the critical variance of the disorder are generally found to be non-monotonic functions of the correlation time of the interactions. We also find that a re-entrant phase transition can take place when this correlation time is varied.