On Products of Random Matrices

TL;DR

This paper introduces a novel class of matrix models based on dessins d'enfant, connecting combinatorial graph structures on surfaces with random matrix theory and algebraic quantities like Hurwitz numbers.

Contribution

It develops a new framework linking graph embeddings on surfaces with multimatrix models and algebraic invariants, expanding the scope of random matrix applications.

Findings

Expressions for integrals in terms of spectra of stars

Connections established with Hurwitz numbers and group characters

New models bridging combinatorics, geometry, and algebra

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.