Two coupled matrices: eigenvalue correlations and spacing functions

Abstract

For two n*n Hermitian matrices A and B, with joint probability density proportional to exp(- tr(U(A)+V(B)+2cAB)), where U and V are polynomials, a method is given to calculate all correlation, cluster and spacing functions of the eigenvalues of either one or both matrices. The method relies on the introduction of two sets of bi-orthogonal polynomials with non-local weights. In a linear chain of coupled matrices, if one looks for the statistical properties of the eigenvalues of only one matrix (two matrices), situated anywhere in the chain, then we can proceed as a one-matrix (two-matrices) problem.