# Determinants of Random Block Hankel Matrices

## TL;DR

This paper investigates the asymptotic behavior of Hankel determinants derived from random matrix measures, establishing convergence results and large deviation principles using novel relations in the Jacobi-beta-ensemble.

## Contribution

It introduces new relations between determinants of subblocks of the Jacobi-beta-ensemble, generalizing Bartlett decomposition results and analyzing their impact on Hankel matrix determinants.

## Key findings

- Weak convergence of the Hankel determinant process is established.
- Mod-Gaussian convergence is demonstrated.
- Large and moderate deviation principles are derived.

## Abstract

We consider the moment space $\mathcal{M}^{p}_{2n+1}$ of moments up to the order $2n + 1$ of $p_n\times p_n$ real matrix measures defined on the interval $[0,1]$. The asymptotic properties of the Hankel determinant $\{\log\det (M_{i+j}^{p_n})_{i,j=0,\ldots,\lfloor nt\rfloor}\}_{t\in [0,1]}$ of a uniformly distributed vector $(M_1,\dots ,M_{2n+1})^t\sim\mathcal{U}(\mathcal{M}_{2n+1})$ are studied when the dimension $n$ of the moment space and the size of the matrices $p_n$ converge to infinity. In particular weak convergence of an appropriately centered and standardized version of this process is established. Mod-Gaussian convergence is shown and several large and moderate deviation principles are derived. Our results are based on some new relations between determinants of subblocks of the Jacobi-beta-ensemble,which are of their own interest and generalize Bartlett decomposition-type results for the Jacobi-beta-ensemble from the literature.