Non-Hermitian β-Ensembles

Abstract

 Non-Hermitian random matrix theory has long been a cornerstone of mathematical physics. A pivotal contribution came from Ginibre’s seminal work [63], which introduced the joint probability density function (j.p.d.f.) of the eigenvalues of matrices with independent complex normal entries. This ensemble, known as Ginibre’s Unitary Ensemble (GinUE), models a two-dimensional system of charged particles in equilibrium under logarithmic repulsion with a quadratic confining potential at inverse temperature β = 2.

 In the Hermitian setting, the β-Hermite ensemble, introduced by Dumitriu and Edelman [40], extended the classical Gaussian orthogonal, unitary, and symplectic ensembles, corresponding to β = 1,2,4, to arbitrary β ∈ R+, thus moving beyond Dyson’s threefold way [47]. This was achieved by constructing real tridiagonal random matrix models whose j.p.d.f. of the eigenvalues mirrors that of the Gaussian ensembles. Building on this foundation, a natural question arises: can analogous complex tridiagonal matrix models be constructed for the Ginibre ensemble, thereby extending Dumitriu and Edelman’s work to the complex domain?

 In this thesis, we address this question by introducing the first random matrix models for complex β-ensembles. These matrices are complex tridiagonal, with entries drawn from the complex normal and χ-distributions. Remarkably, while the j.p.d.f. of the eigenvalues retains the Vandermonde determinant raised to the power β, it departs from the form of the Ginibre ensembles by including an additional factor: a multidimensional integral over the space of the eigenvectors.

 In the low-temperature regime (β → ∞), we compute the limiting spectral density by generalising techniques developed for the β-Hermite ensembles. A key result is the explicit computation of the characteristic polynomials for the complex tridiagonal random matrix models, which relate to Hermite and Meixner-Pollaczek polynomials. Using free probability theory, we derive the limiting spectral distributions in the large-n and large-β limits.

Date of Award	17 Jun 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Francesco Mezzadri (Supervisor)