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Solvable families of random block tridiagonal matrices

Brian Rider, Benedek Valkó

TL;DR

This work constructs two solvable families of random block-tridiagonal matrices, \\mathtt{H}_{\\beta, n}(r,s) and \\mathtt{W}_{\\beta, n,m}(r,s), for which the joint eigenvalue densities can be computed explicitly in select parameter regimes, revealing interactions beyond the standard log-gas. By biasing Dumitriu–Edelman-type ensembles and exploiting a fundamental spectral identity involving a matrix \\mathbf{M}(\\lambda,\\mathbf{Q}), the authors express the biased densities in closed form (density1 for $\\beta s=2$ and density2 for $r=2,\\beta s=4$) with fully explicit normalizing constants. They develop a Gaussian-reduction framework to evaluate determinant-moment functionals and prove several Vandermonde-related determinantal identities, including Pfaffian and Hafnian connections, which underpin the new edge-limit descriptions. The soft-edge limits of these block ensembles are described both via stochastic Airy operators and coupled matrix-diffusion systems, yielding Airy-\\beta-type processes and confirming classical Tracy–Widom distributions in certain regimes; the hard-edge limits of block Laguerre ensembles lead to generalized Bessel-type processes. The results bridge exact solvability with modern edge asymptotics, expanding the landscape of solvable random-matrix models and linking block-structure to rich edge phenomena.

Abstract

We introduce two families of random tridiagonal block matrices for which the joint eigenvalue distributions can be computed explicitly. These distributions are novel within random matrix theory, and exhibit interactions among eigenvalue coordinates beyond the typical mean-field log-gas type. Leveraging the matrix models, we go on to describe the point process limits at the edges of the spectrum in two ways: through certain random differential operators, and also in terms of coupled systems of diffusions. Along the way we establish several algebraic identities involving sums of Vandermonde determinant products.

Solvable families of random block tridiagonal matrices

TL;DR

This work constructs two solvable families of random block-tridiagonal matrices, \\mathtt{H}_{\\beta, n}(r,s) and \\mathtt{W}_{\\beta, n,m}(r,s), for which the joint eigenvalue densities can be computed explicitly in select parameter regimes, revealing interactions beyond the standard log-gas. By biasing Dumitriu–Edelman-type ensembles and exploiting a fundamental spectral identity involving a matrix \\mathbf{M}(\\lambda,\\mathbf{Q}), the authors express the biased densities in closed form (density1 for and density2 for ) with fully explicit normalizing constants. They develop a Gaussian-reduction framework to evaluate determinant-moment functionals and prove several Vandermonde-related determinantal identities, including Pfaffian and Hafnian connections, which underpin the new edge-limit descriptions. The soft-edge limits of these block ensembles are described both via stochastic Airy operators and coupled matrix-diffusion systems, yielding Airy-\\beta-type processes and confirming classical Tracy–Widom distributions in certain regimes; the hard-edge limits of block Laguerre ensembles lead to generalized Bessel-type processes. The results bridge exact solvability with modern edge asymptotics, expanding the landscape of solvable random-matrix models and linking block-structure to rich edge phenomena.

Abstract

We introduce two families of random tridiagonal block matrices for which the joint eigenvalue distributions can be computed explicitly. These distributions are novel within random matrix theory, and exhibit interactions among eigenvalue coordinates beyond the typical mean-field log-gas type. Leveraging the matrix models, we go on to describe the point process limits at the edges of the spectrum in two ways: through certain random differential operators, and also in terms of coupled systems of diffusions. Along the way we establish several algebraic identities involving sums of Vandermonde determinant products.

Paper Structure

This paper contains 19 sections, 28 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Soft edge asymptotics of $\mathtt{H}_{\beta, n}(r,s)$
  3. Block Laguerre ensembles
  4. Preliminaries
  5. Block Jacobi matrices
  6. Random block matrices
  7. New distributions via biasing
  8. Revisiting the Hermite beta ensemble
  9. Key spectral identity
  10. Proofs of Theorem \ref{['thm:main']} and \ref{['thm:main_W']}
  11. A reduction and a representation
  12. Evaluating $E_{\boldsymbol{Q}} \left[|\det \mathbf{M}({\lambda}, \mathbf{Q})|^2\right]$
  13. Evaluating $E_{\boldsymbol{Q}} \left[|\det \mathbf{M}({\lambda}, \mathbf{Q})|^4\right]$ for $r=2$
  14. Determinantal identities
  15. A conjecture on certain Vandermonde sums
  16. ...and 4 more sections

Key Result

Theorem 1

For $\beta =1$ and $2$, the symmetrized joint eigenvalue density of $\mathtt{H}_{\beta, n}(r,s)$ can be computed explicitly in the following cases: and It further holds that with $a_{n,\beta,r,s}=-\frac{\beta}{4} n r (n (r+s)+s)+\left(\tfrac{\beta}{4}-\tfrac{1}{2}\right){nr}$ for all $n$, $\beta = 1$ and $2$, and combinations of $r$ and $s$ in density1 and density2.

Theorems & Definitions (58)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Conjecture 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Definition 9
  • Theorem 10: Spike2
  • ...and 48 more