Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix

Abstract: The universality of correlation functions of eigenvalues of large random matrices has been observed in various physical systems, and proved in some particular cases, as the hermitian one-matrix model with polynomial potential. Here, we consider the more difficult case of a unidimensional chain of matrices with first neighbour couplings and polynomial potentials. An asymptotic expression of the orthogonal polynomials allows to find new results for the correlations of eigenvalues of different matrices of the chain. Eventually, we consider the limit of the infinite chain of matrices, which can be interpreted as a time dependent one-matrix model, and give the correlation functions of eigenvalues at different times.