Products of random Hermitian matrices and brickwork Hurwitz numbers. Products of normal matrices

TL;DR

This paper explores the connection between products of random Hermitian and normal matrices and Hurwitz numbers, revealing new relationships in algebraic geometry and random matrix theory.

Contribution

It establishes a novel link between matrix products and Hurwitz numbers, extending the understanding of ramified coverings in algebraic geometry.

Findings

Derived formulas relating matrix products to Hurwitz numbers

Identified specific ramification types associated with matrix products

Extended the theory to include normal matrices

Abstract

We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified coverings of the Riemann sphere with arbitrary ramification type over $0$ and $\infty$ and ramifications related to the partition $(2^k)$ (``brickworks'' - involution without fixed points) elsewhere. Products of normal random matrices are also considered.