The Anti-Symmetric GUE Minor Process

TL;DR

This paper connects a multi-component particle system from tiling problems to the eigenvalue distribution of anti-symmetric GUE minors, deriving explicit correlation kernels and their scaling limits, including new kernels at the hard edge.

Contribution

It introduces a novel particle system model linked to anti-symmetric GUE minors and derives explicit determinantal correlation kernels with new edge behavior results.

Findings

Joint eigenvalue PDF matches the particle system in a scaling limit.

Explicit correlation kernel expressed via Hermite polynomials.

Identifies new kernel at the hard edge.

Abstract

Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species $j$ consists of $\lfloor j/2 \rfloor$ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge.