# Random Matrices and Integrable Systems

Wednesday 16th JuneFriday 18th June 2021

We’re very excited to welcome Random Matrices and Integrable Systems online event between 16-18th June 2021.

The past decade has seen enormous progress in understanding the behaviour of large random matrices and interacting particle systems.

A large variety of complementary methods have emerged to study these systems, and in particular methods originating from integrable systems.

This workshop is the first of a series of workshops organized within the PIICQ network “Integrable probability, classical and quantum integrability”.

This event has now passed. Presentations can be viewed via the following link.

### Confirmed speakers

#### Giulio Ruzza (Université’ Catholique de Louvain)

Title: Airy kernel determinants, Korteweg-de Vries equation, and nonlocal Painlevé equations

Abstract: I will report on a joint work with Mattia Cafasso and Tom Claeys. We consider Fredholm determinants arising from conditional expectations of the Airy point process and show by a 2×2 Riemann-Hilbert method that they solve the Korteweg-de Vries (KdV) equation. This approach also allows

*to give a streamlined proof that the associated wave function satisfies a nonlocal version of the Painlevé II equation, originally discovered in the context of the Kardar-Parisi-Zhang narrow wedge solution by Amir, Corwin, and Quastel, and

*to study the initial data of the corresponding KdV solutions, the latter being unbounded and thus not belonging to the classes of KdV solutions usually studied in the literature.

#### Karol Kozlowski (ENS Lyon)

Title: Convergence of the form factor series in the Sinh-Gordon quantum field theory in 1+1 dimensions

Abstract: Fredholm determinants of id + K with K an integrable integral operator arise, in particular, as representations for the correlation functions in the simplest instance of quantum integrable models: those explicitly equivalent to a model of free, viz. non-interacting, fermions. Various series of multiple integrals, namely those whose n th summand is given by a n-fold integral, have been obtained for the more complex, truly interacting, quantum integrable models. For numerous reasons, these series provide one with a new class of special functions which naturally generalise the one subordinate to Fredholm determinants of integrable integral operators. However, the status of these representations is still conjectural in that the question of convergence of such series is basically open. The resolution of this convergence problem constitutes thus a necessary step so as to put the theory of correlation functions in truly interacting quantum intgerable models on rigorous grounds but also so as to push further the boarders of special function theory. In this talk, I will report on the recent progress I achieved on the mentioned convergence problem in the case of the simplest non-trivial truly interacting quantum integrable model: the 1+1 dimensional Sinh-Gordon quantum field theory. After reviewing the physical origin of the problem, I will describe the technique that allows one to prove the convergence of the class of series of multiple integral which represent the correlation functions in this model. The proof amounts to obtaining a sufficiently sharp estimate on the leading large-n behaviour of the n-fold integral arising in this context. This appeared possible by refining some of the techniques that were fruitful in the analysis of the large-n behaviour of integrals over the spectrum of n × n random Hermitian matrices.

#### Sofia Taricone (Angers)

Title: Higher order finite temperature Airy kernels and an integro-differential Painlev´e II hierarchy.

Abstract: In this talk we will study Fredholm determinants of a finite temperature version of the higher order Airy kernels that recently appeared in statistical mechanics literature. The main result is an expression of these Fredholm determinants in terms of distinguished solutions of an integrodifferential Painlev´e II hierarchy. Our result generalizes the case n = 1, already studied by Amir Corwin and Quastel some years ago for a special choice of the weight function. This latter can be seen as a generalization of the well known formula connecting the Tracy-Widom distribution for GUE and the Hastings-McLeod solution of the Pailev´e II equation. The proof of our result, for generic n, relies on the study of some operator-valued Riemann-Hilbert problem that builds up the bridge between the description of the Fredholm determinants and the derivation of a Lax pair for this new integro-differential hierarchy. The talk is based on a joint work with Thomas Bothner and Mattia Cafasso, avaiable at https://arxiv.org/pdf/2101. 03557.pdf.

#### Jérémie Bouttier (CEA Saclay/ENS de Lyon)

Title: Multicritical Schur measures

Abstract: Schur measures are random integer partitions, that map to determinantal point processes. We explain how to construct such measures whose edge behavior (asymptotic distribution of the largest parts) is governed by a higher-order analogue of the usual Airy ensemble/Tracy-Widom GUE distribution. This “multicritical” analogue was previously encountered in models of fermions in non-harmonic traps, considered by Le Doussal, Majumdar and Schehr. These authors noted a coincidental connection with unitary random matrix models, which our construction explains via an exact mapping.

#### Mattia Cafasso (Angers)

Title: The generating function for the higher order Airy point processes and a vector-valued Painlevé II hierarchy.

Abstract: During this talk I will present an ongoing joint work with Sofia Tarricone, in which we show that the Fredholm determinants associated to the higher order Airy kernels with several discontinuities admit a Tracy-Widom formula in terms of a (new?) vector-valued Painlevé II hierarchy. The algebraic structure of this hierarchy is quite similar to the one associated to the integro-differential Painlevé II hierarchy we studied with Thomas Bothner and Sofia Tarricone, and which will be illustrated by Sofia Tarricone during her talk.

#### Guido Mazzuca (SISSA, Trieste)

Title: Correlation functions for short-range oscillator

Abstract: Correlation functions are related to transport properties of the materials. We study the correlation functions of a chain of harmonic oscillators with short range interactions when the data is sampled according to the Gibbs measure. By introducing a local generalization of spacing that reduces to relative distance between neighboring oscillators when the potential has only nearest neighbor interactions, we are able to calculate the correlation functions.
Such functions display rapid oscillations that travel along the chain.
We determine the scaling in time of the fastest traveling peaks that are asymptotically described by the Airy function.

This talk is main based on a joint work with T. Grava, T. Kriecherbauer, and K.T-R. McLaughling. https://link.springer.com/article/10.1007%2Fs10955-021-02735-z

#### Jinho Baik (University of Michigan Ann Arbor)

Title: Multi-time distributions of the KPZ fixed point

Abstract: The KPZ fixed point is a two-dimensional random field which is expected to be the universal limit of many random growth models and interacting particle systems that belong to the KPZ universality class. The one-point distribution of the KPZ fixed point for the step initial condition is the Tracy-Widom distribution and the one-dimensional slice in the spatial direction is distributed as the Airy process.

Recently the multi-time, multi-position distribution functions are computed explicitly by Johansson and Rahman and also independently by Liu. We discuss these distribution functions and the associated integrable differential equations.

#### Theo Assiotis (University of Edinburgh)

Abstract: Neretin constructed an analogue of the Hua (or Cauchy type) measures on infinite p-adic matrices. Later Bufetov and Qiu classified the ergodic measures on infinite p-adic matrices for the natural action of $GL(\infty,\mathbb{Z}_p) \times GL(\infty,\mathbb{Z}_p)$. In this talk I will describe explicitly how the Hua measures of Neretin decompose into the ergodic measures of Bufetov and Qiu. This involves the scaling limit of an analogue of the singular values in this setting and some random partitions.

#### Laure Dumaz (ENS Paris)

Title: Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization.

Abstract: We consider the 1-dimensional continuous Schrodinger operator – d^2/d^x^2 + B’(x) on an interval of size L where the potential B’ is a white noise. We study the entire spectrum of this operator in the large L limit. We prove the joint convergence of the eigenvalues and of the eigenvectors and describe the limiting shape of the eigenvectors for all energies. When the energy is much smaller than L, we find that we are in the localized phase and the eigenvalues are distributed as a Poisson point process. The transition towards delocalization holds for large eigenvalues of order L. In this regime, we show the convergence at the level of operators. The limiting operator is acting on R^2-valued functions and is of the form “J \partial_t + 2*2 noise matrix” (where J is the matrix ((0, -1)(1, 0))), a form which already appeared as a conjecture by Edelman Sutton (2006) for limiting random matrices. Joint works with Cyril Labbé.

#### Nick Simm (University of Sussex)

Talk 1 Title: Characteristic polynomials of non-Hermitian matrices, dualities and Painleve transcendents

Abstract 1: I will discuss recent results on moments of characteristic polynomials of non-Hermitian random matrices. Various duality formulae are obtained and used to make a connection to Painlev’e transcendents. This allows us to conjecture the asymptotics of the partition function in more complicated models, such as the critical lemniscate. This is joint work with Alfredo Deano (Madrid).

Talk 2 Title: Secular coefficients of characteristic polynomials and holomorphic multiplicative chaos

Abstract 2: I will discuss the secular coefficients of random unitary matrices, a problem first investigated by Diaconis and Gamburd in 2004. Studying the C\beta E version, we obtain asymptotic limiting distributions for the coefficients and study their tightness. This is done by a link with Gaussian multiplicative chaos and our work motivates the introduction of a holomorphic counterpart of the chaos. This is joint work with Joseph Najnudel (Bristol) and Elliot Paquette (McGill).

Talk 3 Title: Real eigenvalues of real random matrices: new results for product ensembles

Abstract 3: Real eigenvalues of random matrices (such as real Ginibre matrices) were first investigated by Edelman et al. (1994). I will discuss two particular models that have been of recent interest, namely products of independent real Ginibre matrices and products of independent truncated orthogonal random matrices. The latter model concerns a recent collaboration with Alex Little (Bristol) and Francesco Mezzadri (Bristol).

#### Tom Claeys (Université' Catholique de Louvain)

Title: Airy kernel determinants, the KdV equation, and conditional point processes

Abstract: A family of unbounded solutions to the Korteweg-de Vries equation arose recently in the context of the Kardar-Parisi-Zhang equation. These solutions can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and are also related to an integro-differential Painlevé II equation. In my talk, I will will give an overview of these developments, and I will discuss in more detail the initial data of this family of solutions and the connection with conditional Airy point processes.

#### Neil O’Connell (University College, Dublin)

Title: Interacting diffusions on positive definite matrices

Abstract: We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

#### Jon Keating (University of Oxford)

Title: Moments of moments and Gaussian Multiplicative Chaos

Abstract: I will discuss connections between the moments of moments of the characteristic polynomials of random unitary (CUE) matrices and Gaussian Multiplicative Chaos, focusing in particular on the critical-subcritical case.

#### Thomas Bothner (University of Bristol)

Title: A Riemann-Hilbert approach to Fredholm determinants of integral Hankel composition operators

Abstract: These three talks will highlight a novel way of characterizing Fredholm determinants of Hankel composition operators via Riemann-Hilbert problems

#### Emilia Alvarez (University of Bristol)

Title: Moments of the logarithmic derivative of characteristic polynomials from SO(N) and USp(2N)

Abstract: I will discuss work with Nina Snaith on asymptotics of moments of the logarithmic derivative of characteristic polynomials of orthogonal SO(N) and symplectic USp(2N) random matrices, evaluated near the point 1. The leading order behaviour in this regime as N tends to infinity is governed by the likelihood that the matrices in each ensemble have an eigenvalue at or near the point 1. These results follow work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, where they compute these asymptotics in the case of unitary random matrices.

#### Bhargavi Jonnadula (University of Bristol)

Title: Symmetric function theory and moments of characteristic polynomials

Abstract: Symmetric function theory has proven to play a crucial role in random matrix theory to study fundamental quantities such as the joint moments of traces and correlations of the characteristic polynomial. For Hermitian ensembles, we compute exact formulae for moments of characteristic polynomials using symmetric function theory, which further allows us to compute large matrix asymptotics. We comment on the universal properties of the moments of characteristic polynomials for the Gaussian unitary ensemble and how to recover the semi-circle law in the large matrix limit.

This is joint work with Jon Keating and Francesco Mezzadri.

#### Massimo Gisonni (SISSA, Trieste)

Title: Correlators of the Jacobi Unitary Ensemble

Abstract: The Jacobi Unitary Ensemble (JUE) arises as the Beta distribution over Hermitian matrices. In this talk, we focus on the computation and interpretation of its correlators. Respectively, on one side the Matrix Resolvent method, introduced by Bertola, Di, Dubrovin, provides closed formulae for the cumulant functions of the JUE. On the other, work of Harnad et al. on KP-hypergeometric tau functions allows us to express their topological expansion in terms of triple monotone Hurwitz numbers.

This talk is based on joint work with T.Grava and G.Ruzza.

### Programme

Wednesday 16 June, 2021

08:55 – 09:00 – Welcome

09:00 – 09:50 – Nick Simm

10:00 – 10:50 – Thomas Bothner

11:00 – 11:30 – Break

11:30 – 12:00 – Bhargavi Jonnadula

12:10 – 12:40 – Emilia Alvarez

12:50 – 15:30 – Lunch

15:30 – 16:00 – Laure Dumaz

16:10 – 16:40 – Theo Assiotis

16:50 – 17:10 – Break

17:10 – 17:40 – Karol Kozlowski

17:50 – 18:20 – Guido Mazzuca

Thursday 17 June, 2021

08:55 – 09:00 – Welcome

09:00 – 09:50 – Nick Simm

10:00 – 10:50 – Thomas Bothner

11:00 – 11:30 – Break

11:30 – 12:00 – Jon Keating

12:10 – 13:30 – Lunch

13:30 – 14:00 – Neil O’Connell

14:10 – 14:40 – Jérémie Bouttier

14:50 – 15:30 – Break

15:30 – 16:00 – Massimo Gisonni

16:10 – 16:40 – Jinho Baik

Friday 18 June, 2021

08:55 – 09:00 – Welcome

09:00 – 09:50 – Nick Simm

10:00 – 10:50 – Thomas Bothner

11:00 – 11:30 – Break

11:30 – 12:00 – Mattia Cafasso

12:10 – 13:30 – Lunch

13:30 – 14:00 – Sofia Tarricone

14:10 – 14:40 – Giulio Ruzza

14:50 – 15:30 – Break

15:30 – 16:00 – Tom Claeys