Physics (Internal)

Eigenvalue distributions of the correlation matrix: A sensitive indicator for phase transitions?

This event is part of the Biophysics/Condensed Matter Seminar Series.

Under a long time project to discover early warning signals of abrupt changes in complex systems, we have undertaken to study the effect on the spectrum of the correlation matrix of synthetic data obtained from dynamical models. I will present analytic results for equilibrium states at phase transitions for discrete models with power-law decay in an arbitrary number of dimensions supported by numerics that include studies of short time series. We find that power-law behavior in space carries over to the spectrum with a power we give explicitly. This behavior can be traced also if we have many short time series. Going beyond equilibrium dynamics we study the non-equilibrium steady states corresponding to various phases in the totally asymmetric simple exclusion process, a soluble stochastic model of a one lane one way street with given and probabilities of entrance and exit. In particular we consider the transition limit between the high-density and low-density conditions of the entrance and exit probabilities, known to carry a first order phase transition. Again we find a power law, although there is no power law in the exactly know space correlations. At the other critical lines and in the constant current regions we find unusual behavior for the lowest eigenvalues, as yet not understood. In the high and low density regions respectively the Marchenko Pastur distribution, corresponding to white-noise time-series is found for the eigenvalues to excellent numerical accuracy.