On Products of Random Matrices

Entropy (Basel). 2020 Aug 31;22(9):972. doi: 10.3390/e22090972.

Abstract

We introduce a family of models, which we name matrix models associated with children's drawings-the so-called dessin d'enfant. Dessins d'enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the "spectrum of stars". The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory.

Keywords: Hurwitz number; Schur polynomial; generalized hypergeometric functions; integrable systems; matrix models; products of random matrices; random complex and random unitary matrices.