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

TL;DR

This paper introduces a new Hermitian matrix ensemble that models non-intersecting Brownian bridges with arbitrary endpoint multiplicities, providing explicit realizations and exact finite-n results.

Contribution

It constructs a unitarily invariant matrix model matching the Karlin--McGregor law for non-intersecting Brownian bridges and derives several exact finite-n consequences.

Findings

Eigenvalue law matches Karlin--McGregor law for non-intersecting Brownian bridges.

Partition function reduces to a single HCIZ integral with explicit time dependence.

Spectral equivalence but angular statistics differ between the constructed ensemble and Gaussian external-field ensemble.

Abstract

We construct a unitarily invariant Hermitian matrix ensemble whose fixed-time eigenvalue law coincides with the Karlin--McGregor law for non-intersecting Brownian bridges with arbitrary finite multiplicities at both endpoints. This provides an explicit matrix-ensemble realization of the known mixed-type multiple orthogonal polynomial and Riemann--Hilbert description of the general multi-start/multi-end problem. We then derive several exact finite-$n$ consequences of this construction. These include a path-space lift as an orbital Hermitian Brownian bridge and a reduction of the partition function to a single compact HCIZ integral with explicit $t$-dependence. We also compare the one-sided reduction with the Gaussian external-field ensemble, showing that, although the two ensembles are spectrally equivalent, their angular statistics are different. Finally, we derive fixed-time Schwinger--Dyson identities and associated resolvent relations for the dressed ensemble.