Workshop on Random Matrices and Integrability

Mini courses:

• Speaker: Theo Assisotis (University of Edinburgh)

  Title: Moments of characteristic polynomials of random matrices

  Abstract: In this lecture series I will be talking about the asymptotics of various generalised moments of characteristic polynomials of random matrices from the classical compact groups. The two model problems we will consider are (1) the asymptotics of joint moments of characteristic polynomials with their derivatives and (2) the asymptotics of so-called moments of moments. The first lecture will give some motivation for the area and overview of the problems and results. The second lecture will then focus on the problem of joint moments and corresponding techniques used. The third lecture will focus on the problem of moments of moments and the corresponding techniques to attack it. 

  These talks are based on several papers over the past years which are joint work with B. Bedert, M. A. Gunes, J. P. Keating, J. Najnudel, A. Soor, J. Warren and F. Wei.

   

• Speaker: Joseph Najnudel (University of Bristol)
   
  Title: Characteristic polynomial of circular ensembles and holomorphic multiplicative chaos. 
   
  Abstract: A consequence of results by Diaconis and Shahshahani, generalized by Jiang and Matsumoto, is the fact that the characteristic polynomial of the Circular Unitary Ensemble, and more generally the Circular Beta Ensemble, converges to the exponential of a Gaussian holomorphic function, with very explicit covariance structure. In joint articles with Atherfold, Paquette and Simm, we study limit theorems on the coefficients of this exponential of Gaussian series, called Holomorphic multiplicative chaos, because of its direct link with the Gaussian multiplicative chaos. We obtain non-gaussian limiting distributions, related to the distribution of the total mass of the Gaussian multiplicative chaos. Our results on the holomorphic multiplicative chaos have consequences on the distribution of the coefficients of characteristic polynomials. In particular, we show that the middle coefficient of the Circular Unitary Ensemble tends to zero in probability, answering a question by Diaconis and Gamburd. Our work is also related to partial sums of random multiplicative functions in analytic number theory.
   
• Speaker: Mo Dick Wong (Hong Kong University)
   
  Title: Introduction to Gaussian multiplicative chaos
   
  Abstract: Gaussian multiplicative chaos (GMC) is a family of random measures originally motivated by the Kolmogorov–Obukhov–Mandelbrot model of turbulence. First constructed rigorously by Kahane in 1985, it has since become an important object in mathematical physics, with applications to areas such as Liouville conformal field theory, random matrices and number theory. These lectures will give an introduction to the basic theory of GMC, beginning with Berestycki’s elementary construction of subcritical chaos, before turning to its multifractal properties and distributional behaviour. We will also see how exponential functionals of Brownian motion provide a useful toy model for understanding questions such as the universality of right-tail probability asymptotics. Time permitting, we will discuss further topics and applications.

Research talks: 

• Speaker: Tamara Grava (SISSA and University of Bristol)
   
  Title: From beta-ensembles to generalized Gibbs ensembles.
   
  Abstract: I will consider the generalized Gibbs ensembles of discrete integrable systems. These ensembles  represent several classes  of  random matrix models.  In particular the generalized Gibbs ensembles of the Toda lattice and the Ablowitz lattice are related to  the  beta-ensembles and  insights  into their behaviour can be obtained by exploiting this proximity. I will present an  overview of  recent results in the field.
   
• Speaker: Jad Hamdan (University of Oxford)
   
  Title: The phases of Gaussian multiplicative chaos, and some number-theoretic implications
   
  Abstract: I will give an introduction to the many phases of Gaussian multiplicative chaos, explaining the different normalisation schemes required in the critical and supercritical regimes. This will be followed by a discussion of how these arise in recent results in number theory and random matrix theory, downstream of conjectures of Fyodorov, Hiary, and Keating. 

• Speaker: Nick Simm (University of Sussex)
   
  Title: Characteristic polynomials and planar orthogonality in non-Hermitian ensembles
   
  Abstract: I will discuss recent developments concerning moments of characteristic polynomials in non-Hermitian random matrix ensembles, focusing on truncations of Haar-distributed unitary matrices. The corresponding characteristic polynomial moments exhibit a rich structure, including connections with Painlevé transcendents. Depending on the regime considered, Painlevé IV, V or VI can occur. These moments are also naturally related to a class of orthogonal polynomials in the complex plane.
   
  For these polynomials, planar orthogonality can be reformulated as a Riemann–Hilbert problem on suitable contours. I will describe some aspects of the resulting asymptotic analysis, based on the Deift–Zhou steepest descent method, and explain how it yields strong asymptotics for both the orthogonal polynomials and the characteristic polynomial moments. This is based on joint work with Alfredo Deaño and, more recently, with Kenneth McLaughlin and Leslie Molag.