Organizers : Giuseppe Orsatti (UCLouvain)
2024-2025
Tuesday, November 19 at 2:00 pm, room b.203
Carla Da Silva (University of São Paulo-Larema)
Title : Random Matrix theory and deformations of the Painlevé I Kernel
Abstract : Applications of Random Matrix Theory can be found in almost every research area. From nuclear resonance to machine learning and even natural phenomena are some examples of systems whose behavior is predicted by eigenvalues in random matrix ensembles. In unitary ensembles the eigenvalues constitute a Determinantal Point Process and, consequently, the relevant statistics can be obtained through the study of the correspondent reproducing kernel. In this context, it is well known that when the equilibrium measure related to the probability measure on the space of matrices has a “soft edge” (behaves as x1/2), the associated kernel converges to the Airy kernel. And, as stated in [1], as the edge becomes “softer” (behavior of order x5/2, x9/2 and so on), the limit kernel is given by means of solutions to the Painlevé I hierarchy. The present seminar is divided into two parts. Firstly, we present some applications and standard techniques that motivates the studies of kernels arising in RMT as well as deformations of such kernels. Next, we present briefly some original results for a deformed higher-order Painlevé I kernel.
Tuesday, October 29 at 2:00 pm, room b.203
Christophe Charlier (UCLouvain)
Title: Hole probabilities and balayage of measures for planar Coulomb gases
Abstract: Coulomb gases consist of n particles repelling each other via the 2d Coulomb law and subject to the presence of an external potential. In this talk, I will discuss recent results on the probability that a given subset of the plane is free from particles when n is large. I will also talk about the most likely point configurations ("far from equilibrium") that produce such holes, which are described in terms of balayage measures.
Tuesday, October 8 at 2:00 pm, room b.203
Prof. Arno Kuijlaars (KU Leuven)
Title : Asymptotic analysis of matrix valued orthogonal polynomials from 3x3 doubly periodic random tilings of an hexagon
Abstract: I will discuss a class of matrix valued orthogonal polynomials (MVOP) that arise from random tilings of a hexagon with doubly periodic weightings.The matrix valued orthogonality is closely connected to scalar orthogonality on a Harnack curve, which is an algebraic curve with very special properties. The zeros of the scalar orthogonal functions distributed themselves along a contour on the Harnack curve. I will explain that the weak limit of the normalized zero counting measures is an equilibrium measure for a bipolar Green’s energy in an external field. This measure is a main ingredient for the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem.
The talk will be followed by a Junior Reading Group on Orthogonal Polynomials and Random Matrices (from 3.15 pm to 4.15 pm)
2023-2024
Tuesday, April 16 at 11:00, room b.203:
Matthew Mitchell (University of Michigan)
Title : The Small-Dispersion Limit for the Intermediate Long Wave Equation
Abstract: The Intermediate Long Wave (ILW) equation is an integrable nonlinear evolution equation which arises as an asymptotic (long-wavelength, small-amplitude) model for internal gravitational waves between two fluid layers, one shallow and the other of depth $\delta > 0$. In the limits of $\delta \to 0$ and $\delta \to \infty$, the ILW equation formally reduces to the Korteweg-de Vries (KdV) equation and the Benjamin-Ono (BO) equation, respectively. For both of these limiting equations, the small dispersion limit for positive, "bell-shaped" initial data has been studied rigorously using their respective Inverse Scattering Transforms (IST), while the problem remains open for the ILW equation. In this talk, we will review the formal IST for the ILW equation, conduct a Wentzel–Kramers–Brillouin-type analysis to generate asymptotic spectral data for a given "bell-shaped" initial condition $u_0$, and prove weak convergence of the solution associated to this spectral data via the IST for small time.
Tuesday, March 19 at 11:00, room b.203:
Aron Wennman (KU Leuven)
Title: A PDE approach to polynomial Bergman kernels and orthogonal polynomials for weighted area measures
Abstract: In this talk I plan to survey some (old and new) results about orthogonal polynomials and related polynomial reproducing kernel functions, defined with respect to weighted area measure on the plane. These objects appear naturally in connection with the 2D Coulomb gas at a particular inverse temperature, and they have recently been used to obtain several universality results for this model.
I plan to describe a number of equivalent but distinct characterizations of planar orthogonality, each offering its own inroad to study large-degree asymptotics. I will then focus on one PDE based approach, which shares a resemblance with the Riemann-Hilbert approach to orthogonal polynomials on the real line. One of the main results is a new simple globally valid asymptotic formula for the so-called one-point function.
The talk is based on joint projects with Giandinoto and with Hedenmalm, respectively.
Tuesday, February 27 at 11:00, room b.203 :
Christian Hagendorf (UCLouvain)
Title: Nearest-neighbour correlation functions for the supersymmetric XYZ spin chain and Painlevé VI
Abstract: The exact computation of correlation functions is one of the key challenges in the theory of interacting quantum systems. In this talk, we present exact results for the nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of Painlevé VI tau functions. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian. This talk is based on joint work with Hjalmar Rosengren.
Tuesday, January 9 2024 at 14:00, room b.203 :
Philippe Moreillon (University of Geneva)
Title: Disk counting statistics of the eigenvalues of truncated unitary matrices
Abstract: In this talk, I will present new results on the disk counting statistics of the eigenvalues of truncated unitary matrices. More specifically, if T is the upper-left submatrix of a Haar distributed unitary matrix of size (n+k) x (n+k), we prove that as n tends to infinity with k fixed, the associated moment generating function verifies asymptotics of the form exp(C_1 n + C_2 + o(1)), where the constants C_1 and C_2 are given in terms of the incomplete Gamma function. Our proof relies on the uniform asymptotics of the incomplete Beta function. This talk is based on a joint work with Christophe Charlier and Yacin Ameur.
Tuesday, December 12 at 14:00, room b.203 :
Sonja Hohloch (University of Antwerp)
Title: On low dimensional integrable Hamiltonian systems with S^1-symmetry
Abstract: This talk gives an introduction and overview of recent developments around so-called hypersemitoric systems. These are two degree of freedom integrable Hamiltonian systems on 4-dimensional compact symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian S^1-action.
Tuesday, November 28 at 14:30 room b.203
Giuseppe Orsatti (SISSA, Italy) (14:30-15:30)
Title: Integrable Operators, dbar problems, Kp and NLS hierarchy.
Abstract: In the theory of Integrable Operators, the main object of study is their Fredholm determinant, which in some scenarios represents a $\tau$-function of Integrable Systems. The works of Its, Izergin, Korepin and Slavnov (IIKS) unveiled a deep connection between the existence of the resolvent of integrable operators acting on curves $\Sigma \subset \mathbb{C}$ and the solution of a Riemann-Hilbert problem. In this talk we enlarge the IIKS theory also for Hilbert-Schmidt Integrable Operators acting on domains $\mathscr{D} \subset \mathbb{C}$. For this class of operators, the existence of the resolvent is given by the solution of a dbar problem. The existence of the resolvent assure the non vanishing of the Hilbert-Carleman determinant of the operator. Finally we show that when the operator has a particular dependence on parameters that we call times, the Hilbert-Carleman determinant represents a $\tau$-function of the Kadomtsev-Petviashvili equation or the nonlinear Schrödinger equation. In particular in this latter case such kind of operators naturally arise as an infinite limit of a $N$-soliton solution.
This talk is based on a joint work with Prof. Grava (University of Bristol) and Prof. Bertola (Concordia University).
Sampad Lahiry (KU Leuven) (15:30 - 16:30)
Title: Orthogonal polynomials in the normal random matrix model with a potential having two logarithmic singularities
Abstract: We consider the orthogonal polynomials $P_{n,N}(z)$ with respect to the weight $$|z^2+a^2|^{Nc}e^{-N|z|^2}dA(z)$$ in the whole complex plane. We obtain strong asymptotic and the limiting zero counting measure (motherbody) of the orthogonal polynomials in the scaling limit $n,N\rightarrow \infty$ such that $\lim \frac{n}{N}=t$. We also find the two dimensional equilibrium measure (droplet) associated with the eigenvalues in the corresponding normal matrix model. Our method relies on observing that the planar orthogonal polynomials are a part of Type I multiple orthogonal polynomial, and this allows us to do steepest descent analysis of the associated Riemann Hilbert problem. This is a joint work with Mario Kieburg and Arno Kuijlaars.
Tuesday, November 21st,CYCL 09A :
Tom Claeys (14:00-15:00) (UCLouvain)
Title: Polymer partition functions and biorthogonal measures
Abstract: The Log Gamma polymer, the O'Connell-Yor polymer and the mixed polymer are special cases of polymer models whose partition functions admit exact expressions as Fredholm determinants. I will explain that these Fredholm determinants are naturally connected to a class of biorthogonal measures. Several well-known random matrix ensembles belong to this class of biorthogonal measures and can be seen as limiting cases of polymer models. I will also discuss possible implications of our results for asymptotic analysis. The talk will be based on joint work in progress with Mattia Cafasso.
Julian Mauersberger (15:00-16:00) (UCLouvain)
Title: Asymptotics for Fredholm determinants related to the log-Gamma polymer
Abstract: In this talk I present ongoing work on Fredholm determinants related to biorthogonal measures characterizing the log-Gamma polymer and similar models. In particular, we are interested in gap probabilities for such models for which we derive Fredholm determinant expressions in terms of integrable kernels. Via the IIKS procedure these expressions admit Riemann-Hilbert representations which enable asymptotic analysis applying the nonlinear steepest descent method. We carry out the steepest descent method and compute the leading order term of an expansion that conjecturally describes the large deviations for the log-Gamma polymer partition function. This is joint work with T. Claeys.