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Random commuting matrices

John E. McCarthy

TL;DR

This work develops a systematic framework to study random $d$-tuples of $n\times n$ matrices constrained by algebraic relations, with a focus on commuting tuples and their eigenvalue distributions. It extends classical random matrix theory to multi-matrix settings by introducing unitarily invariant ensembles on algebraic sets such as ${\frak C}^d_n$ and its Hermitian and non-Hermitian variants, deriving a multidimensional Ginibre-type formula for eigenvalue densities and establishing large-$n$ limits via equilibrium measures for logarithmic potentials. In the Hermitian case, the eigenvalues of suitably scaled random tuples converge to explicit equilibrium measures supported on balls or spheres, yielding projections that follow semicircular laws for low $d$ and concentration near the origin for higher $d$. In the non-Hermitian regime, the irreducible case admits a structured decomposition that leads to explicit densities in low dimensions (notably $n=2$) and reveals attraction phenomena for $d\ge3$, contrasting with repulsion in the Hermitian setting. The results connect random matrix theory with potential theory, free probability, and multivariable operator theory, and open the door to further explorations of spectral laws under algebraic constraints and their implications for related von Neumann algebras and group-theoretic constructions.

Abstract

We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random variable that takes values in the set of commuting $d$-tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the Hermitian case, we characterize the eigenvalue distribution as $n$ tends to infinity. In the non-Hermitian case, we get a formula that holds if the set is irreducible. We show that there are qualitative differences between the single matrix case and the several commuting matrices case.

Random commuting matrices

TL;DR

This work develops a systematic framework to study random -tuples of matrices constrained by algebraic relations, with a focus on commuting tuples and their eigenvalue distributions. It extends classical random matrix theory to multi-matrix settings by introducing unitarily invariant ensembles on algebraic sets such as and its Hermitian and non-Hermitian variants, deriving a multidimensional Ginibre-type formula for eigenvalue densities and establishing large- limits via equilibrium measures for logarithmic potentials. In the Hermitian case, the eigenvalues of suitably scaled random tuples converge to explicit equilibrium measures supported on balls or spheres, yielding projections that follow semicircular laws for low and concentration near the origin for higher . In the non-Hermitian regime, the irreducible case admits a structured decomposition that leads to explicit densities in low dimensions (notably ) and reveals attraction phenomena for , contrasting with repulsion in the Hermitian setting. The results connect random matrix theory with potential theory, free probability, and multivariable operator theory, and open the door to further explorations of spectral laws under algebraic constraints and their implications for related von Neumann algebras and group-theoretic constructions.

Abstract

We define a random commuting -tuple of -by- matrices to be a random variable that takes values in the set of commuting -tuples and has a distribution that is a rapidly decaying continuous weight on this algebraic set. In the Hermitian case, we characterize the eigenvalue distribution as tends to infinity. In the non-Hermitian case, we get a formula that holds if the set is irreducible. We show that there are qualitative differences between the single matrix case and the several commuting matrices case.
Paper Structure (23 sections, 16 theorems, 147 equations)

This paper contains 23 sections, 16 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. What is a random $d$-tuple of matrices that satisfy algebraic relations?
  4. Why are random tuples of matrices interesting?
  5. Main results
  6. Main results: Hermitian case
  7. Main results: General case
  8. Known results
  9. The Gaussian Unitary Ensemble
  10. Equilibrium measures for the logarithmic potential
  11. Other results
  12. Random commuting Hermitian matrices
  13. Generic $d$-tuples
  14. Ginibre formula
  15. Density of eigenvalues
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $X_n$ be a random variable in ${\mathfrak C}^{d,H}_n$, and let the weights be probabilities given by $w_n(X) \ = \ c_n e^{-\frac{1}{2}{\rm tr}[ \sum_{r=1}^d (X^r)^* X^r]}$. As $n \to \infty$, the eigenvalues of $\frac{1}{\sqrt{n}} X_n$ tend to the following equilibrium distribution on the close

Theorems & Definitions (31)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.4
  • Example 1.5
  • Theorem 2.1
  • Theorem 3.1
  • Definition 4.1
  • Proposition 4.2
  • proof
  • ...and 21 more