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A Two-HCIZ Gaussian Matrix Model for Non-intersecting Brownian Bridges

Maksim Kosmakov

TL;DR

The paper constructs a two-HCIZ dressed Gaussian matrix model for non-intersecting Brownian bridges with multiple starts and ends and proves that, at fixed $n$ and time $t$, its eigenvalue density exactly matches the Karlin–McGregor law. A key advance is the single-HCIZ collapse, which factorizes the partition function into an explicit $t$-dependent prefactor and a single HCIZ integral, identifying the partition function as a $2$D-Toda $\tau$-function in Miwa variables and yielding Virasoro (Schwinger–Dyson) constraints. The work also analyzes reductions: in $(p,q)=(1,1)$ it recovers a centered GUE, while in $(p,q)=(2,1)$ the spectral law matches the external-field ensemble but with Haar-distributed eigenvectors, highlighting distinct angular statistics. Overall, the matrix-model formulation provides a unifying origin for the mixed-type MOP/RHP description and enables exact finite-$n identities and large-$n$ asymptotics, linking spectral data to integrable hierarchies and Virasoro symmetries.

Abstract

We construct a one-matrix model for non-intersecting Brownian bridges with multiple starts and ends: it is a Gaussian Hermitian ensemble 'dressed' by two Harish-Chandra-Itzykson-Zuber (HCIZ) integrals encoding the boundary data. We prove that, at finite $n$ (including confluent multiplicities), its eigenvalue law coincides with the Karlin-McGregor distribution. A structural "single-HCIZ collapse" of the partition function, with an explicit $t$-dependent prefactor, identifies it as a $2$D-Toda $τ$-function in Miwa variables and leads to Virasoro constraints via Schwinger-Dyson equations. In the reduction $(p,q)=(2,1)$, the model matches the external-field ensemble spectrally while exhibiting distinct angular statistics (Haar-distributed eigenvectors). These results provide the matrix-integral origin for the mixed-type multiple orthogonal polynomial/Riemann-Hilbert description and enable direct finite-$n$ identities and large-$n$ asymptotics.

A Two-HCIZ Gaussian Matrix Model for Non-intersecting Brownian Bridges

TL;DR

The paper constructs a two-HCIZ dressed Gaussian matrix model for non-intersecting Brownian bridges with multiple starts and ends and proves that, at fixed and time , its eigenvalue density exactly matches the Karlin–McGregor law. A key advance is the single-HCIZ collapse, which factorizes the partition function into an explicit -dependent prefactor and a single HCIZ integral, identifying the partition function as a D-Toda -function in Miwa variables and yielding Virasoro (Schwinger–Dyson) constraints. The work also analyzes reductions: in it recovers a centered GUE, while in the spectral law matches the external-field ensemble but with Haar-distributed eigenvectors, highlighting distinct angular statistics. Overall, the matrix-model formulation provides a unifying origin for the mixed-type MOP/RHP description and enables exact finite-n$ asymptotics, linking spectral data to integrable hierarchies and Virasoro symmetries.

Abstract

We construct a one-matrix model for non-intersecting Brownian bridges with multiple starts and ends: it is a Gaussian Hermitian ensemble 'dressed' by two Harish-Chandra-Itzykson-Zuber (HCIZ) integrals encoding the boundary data. We prove that, at finite (including confluent multiplicities), its eigenvalue law coincides with the Karlin-McGregor distribution. A structural "single-HCIZ collapse" of the partition function, with an explicit -dependent prefactor, identifies it as a D-Toda -function in Miwa variables and leads to Virasoro constraints via Schwinger-Dyson equations. In the reduction , the model matches the external-field ensemble spectrally while exhibiting distinct angular statistics (Haar-distributed eigenvectors). These results provide the matrix-integral origin for the mixed-type multiple orthogonal polynomial/Riemann-Hilbert description and enable direct finite- identities and large- asymptotics.
Paper Structure (23 sections, 7 theorems, 92 equations)

This paper contains 23 sections, 7 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and the Karlin--McGregor Law
  3. Notation and Setup
  4. The Karlin--McGregor Law for Brownian Bridges
  5. The Two-HCIZ Dressed Ensemble
  6. Main Results
  7. Spectral Equivalence: Proof of \ref{['thm:KM-equivalence']}
  8. Reductions
  9. Case $(p,q)=(1,1)$
  10. Case $(p,q)=(2,1)$
  11. Derivation of the (2,1) Reduction
  12. Comparison to the External-Field Model: Angular vs. Spectral Statistics
  13. Structural Properties: Proofs of \ref{['thm:collapse', 'cor:tflow-main', 'thm:moments']}
  14. Connection to Known Results and Integrable Structures
  15. MOPs, RHP, and the Matrix Ensemble
  16. ...and 8 more sections

Key Result

Proposition 2.1

The joint probability density for the positions $\bm\lambda = (\lambda_1, \dots, \lambda_n)$ of $n$ non-intersecting Brownian bridges at time $t$ is given by In the case of multiplicities in the start or end points, the corresponding determinants are understood as their confluent limits, which involve derivatives with respect to the start/end parameters (see app:confluent-HCIZ).

Theorems & Definitions (16)

  • Proposition 2.1: Karlin--McGregor Formula for NIBBs
  • Definition 2.2: The Two-HCIZ Dressed Ensemble
  • Theorem 3.1: Finite-$n$ Spectral Equivalence
  • Proposition 3.2: Reductions and Angular Statistics
  • Theorem 3.3: Single-HCIZ Collapse
  • Corollary 3.4: Time Flow and Time-Reversal Duality
  • Corollary 3.5: Miwa parametrization and 2D-Toda
  • Theorem 3.6: Finite-$n$ Moments and Angular Statistics
  • proof : Outline of Proofs
  • Remark 5.1: Scalar translations in the reductions
  • ...and 6 more