Diffusions on symmetric spaces of type A III and random matrix theories for rectangular matrices

Isotropic diffusion processes on cosets U(M+N)/U(M)xU(N) and U(M,N)/U(M)xU (N) and their zero-curvature limit are studied from a unified viewpoint. As indicated in our previous works the projection of the Fokker-Planck equation onto the maximal commutative subgroup of these cosets can be described by using the radial part of the Laplace-Beltrami operator. By taking the zero-curvature limit, an integral which is a natural extension of the Itzykson-Zuber integral to the rectangular matrices is explicitly evaluated. The probability density function obtained from the diffusion on U(M,N)/U(M)xU(N) is studied in detail, which can be applied to the quantum transport problem. The explicit expressions for the probability density function in the metallic and insulating regimes are obtained. For the metallic regime the integral representation for the hypergeometric function is used and the results are exact. Furthermore, by using the orthogonal polynomial method n-point correlation functions are obtained exactly for arbitrary n. (author)