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

Ch. Li, A. Yu. Orlov

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.

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

Abstract

We consider products of 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 -fold ramified coverings of the Riemann sphere with arbitrary ramification type over and and ramifications related to the partition (``brickworks'' - involution without fixed points) elsewhere. Products of normal random matrices are also considered.
Paper Structure (9 sections, 7 theorems, 48 equations)

This paper contains 9 sections, 7 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Matrix models (\ref{['new']}), (\ref{["new'"]})
  3. Preliminaries
  4. Matrix models as perturbation series
  5. Brickwork Hurwitz numbers
  6. Products of normal matrices
  7. Tau functions
  8. Discussion
  9. Characteristc map and the Frobenius relation

Key Result

Proposition 2.1

Let $A$ and $B$ be $N\times N$ complex matrices. Then where $d_*U$ is the Haar measure on $\mathbb{U}_N$ and $C_N$ is chosen from the normalization $C_N\int_{\mathbb{U}_N} d_*U=1$ (and in what follows we include it to the definition of $d_*U$) and where $s_\lambda$ is the Schur function labeled by a partition $\lambda$, see Mac.

Theorems & Definitions (10)

  • Proposition 2.1
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.2
  • Corollary 2.1
  • Proposition 2.3
  • Remark 2.2
  • Proposition 3.1
  • Remark 3.1