2 COUPLED MATRICES - EIGENVALUE CORRELATIONS AND SPACING FUNCTIONS

For two n x n Hermitian matrices A and B, with joint probability densi ty proportional to exp[-tr{U(A) + V(B) + 2cAB}], where U and V are pol ynomials, a method is given to calculate all correlation, cluster and spacing functions of the eigenvalues of either one or both matrices. T he method relies on the introduction of two sets of bi-orthogonal poly nomials with non-local weights. In a linear chain of coupled matrices, if one looks for the statistical properties of the eigenvalues of onl y one matrix (two matrices), situated anywhere in the chain, then we c an proceed as a one-matrix (two-matrices) problem.