## Focused Research: Non-Hermitian Random Matrices

16 Feb 2022, by Sponsored events in7 – 11 November 2022

School of Mathematics, University of Bristol, UK

**Organisers: **

Thomas Bothner (Bristol)

Tamara Grava (Bristol)

**Participants: **

Gernot Akemann (Bielefeld)

Thomas Bothner (Bristol)

Tamara Grava (Bristol)

Boris Khoruzhenko (QMUL)

Guido Mazzuca (KTH Stockholm)

Francesco Mezzadri (Bristol)

Sofia Tarricone (LAREMA, Angers/Concordia, Montréal)

Nickolas Simm (Sussex)

**Titles and Abstracts:****Sofia Tarricone** (Leuven): **From the Fredholm determinant of the finite temperature Airy kernel to the Janossy densities of the thinned Airy dpp.**

**Abstract**: The aim of the talk is to show that the m-th Janossy densities of a suitably thinned Airy process are governed by the Schrödinger and (cylindrical) KdV equations; moreover, we prove that the associated wave functions satisfy a system of coupled integro-differential Painlevé II equations. These results generalize the ones already known for the Fredholm determinant of the so called finite temperature Airy kernel (recovered when m=0), that appears in relation with multiple models such as the KPZ equation, non-intersecting fermions at finite temperature with harmonic potential, and it was recently related also to some weak non-Hermiticity limit in the elliptic Ginibre ensemble. The talk is based on ongoing work with T. Claeys, G. Glesner and G. Ruzza.

**Thomas Bothner** (Bristol): **Spacing distributions in the elliptic Ginibre ensemble at weak non-Hermiticity.**

**Abstract**: I will discuss the limiting edge and bulk spacing distributions in the complex elliptic Ginibre ensemble, in the limit of weak non-Hermiticity. This is joint work with Alex Little (Bristol).

**Henry Taylor** (Bristol): **Some interesting properties of the spectra of complex tridiagonal matrices.**

**Abstract**: In this talk we will discuss some interesting features of the eigenvalues of a particular ensemble of complex random tridiagonal matrices. This is work in progress with Francesco Mezzadri.

**Nick Simm ** (Sussex): **Moments of characteristic polynomials of non-Hermitian ensembles.**

**Abstract**: I will review some of the known results about moments of characteristic polynomials of non-Hermitian ensembles. One interesting example is obtained by cutting out the top-left M x M sub-block of a random unitary matrix, sometimes referred to as a truncation. Using the theory of symmetric functions, we can identify the corresponding moments as a certain hypergeometric function of matrix argument. I will discuss some of the benefits this brings, as well as suitable analogues for the orthogonal and symplectic groups. This is joint work with my PhD student Sasha Serebryakov.

**Gernot Akemann** (Bielefeld): **Fermions in a 2D rotating trap and universality in non-Hermitian random matrix theory.**

**Abstract**: It has been shown that the ground state of N noninteracting Fermions in a two-dimensional rotating trap can be described by random matrix theory. The corresponding ensemble is non-Hermitian, the complex Ginibre ensemble which is Gaussian. Its complex eigenvalues enjoy an interpretation as a two-dimensional Coulomb gas and many other applications. We show that the variance of the number of Fermions in a centred disc is universal in the large-N limit for non-Gaussian potentials and only depends on its radius. This universality class also includes a second random matrix ensemble, the symplectic Ginibre ensemble. We exploit the integrable structure of both ensembles to derive compact expressions for the variance valid at finite-N.

This is joint work with Sungsoo Byun and Markus Ebke [arXiv:2206.08815, J Stat Phys to appear

**Boris Khoruzhenko** (QMUL): **Escaping the crowds: extreme eigenvalues and outliers of nearly Hermitian matrices’**

**Abstract**: It is known that in the limit of large matrix dimensions the eigenvalue density of rank-one non-Hermitian additive deviations from the Gaussian Unitary Ensemble undergoes an abrupt restructuring when the magnitude of the deviation reaches a certain threshold beyond which a single eigenvalue outlier appears. I will discuss this restructuring transition, including the width of the critical region about the outlier threshold and the associated characteristic height of extreme eigenvalues (those which are farthest away from the real axis), and offer a quantitative description of gradual separation of the outlier from the rest of the extreme eigenvalues. I will also briefly discuss a related ensemble of random contractions where one can determine, in the limit of large matrix dimensions, the probability distribution of the eigenvalue closest to the origin and show that it interpolates between Gumbel and Frechet laws of extreme value statistics. My talk is based on joint work with Yan Fyodorov and Mihail Poplavskyi (arXiv:2211.00180).

**Guido Mazzucca** (KTH Stockholm): **Integrable systems and random matrices**.

**Abstract**: In this talk, we focus on the relation between random matrix theory and integrable systems. Specifically, we investigate the spectral behaviour of the Lax matrices of some integrable models when the number N of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. In particular, we focus on the families of Itoh–Narita–Bogoyavleskii additive and multiplicative lattices, the focusing Ablowitz–Ladik lattice and the focusing Schur flow. All these integrable systems have a Non–Hermitian Lax matrix, and we derive numerically their density of states that has support in the complex plane. Furthermore, we give some idea on how to use these integrable models to define some new β ensembles with a sparse matrix representation having the eigenvalues on the complex plane.

The talk in mainly based on the recent paper M. Gisonni, T. Grava, G. Gubbiotti, G. M., Discrete integrable systems and random Lax matrices. Forthcoming in Journal of Statistical Physics (2022) DOI:10.1007/s10955-022-03024-z