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

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 $\tau$-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.