Thermodynamics on the spectra of random matrices

Abstract

We show that the spectra of Wishart matrices built from magnetization time series can describe the phase transitions and the critical phenomena of the Potts model with a different number of states. We can statistically determine the transition points, independent of their order, by studying the density of the eigenvalues and corresponding fluctuations. In some way, we establish a relationship between the actual thermodynamics with the spectral thermodynamics described by the eigenvalues. The histogram of correlations between time series interestingly supports our results. In addition, we present an analogy to the study of the spectral properties of the Potts model, considering matrices correlated artificially. For such matrices, the eigenvalues are distributed in two groups that present a gap depending on such correlation.Comment: 10 pages, 11 figure