The GPP seminar is part of the KUL--UCL Classical Analysis and Integrable Systems Seminar, co-organized at KULeuven and UCLouvain.

Information on the talks at KULeuven can be found here

Another adjoint seminar series is the monthly PIICQ Online Seminar on Integrable Probability, Classical and Quantum Integrability, see here

**Organizers** : Alexandre Lazarescu (UCLouvain)

## 2022-2023

**UCL/KUL Integrable systems and Classical analysis Seminar (Sept. 2022– June 2023)**

**Tuesday October 18, 2022, CYCL B203 :**

**Sofia Tarricone** (UC Louvain) [2pm-3pm]

*Toeplitz determinants related to a discrete Painlevé II hierarchy*

Abstract: The aim of this talk is to give a formula describing certain Toeplitz determinants, appearing in some random partitions models, in terms of solutions of a discrete Painlevé II hierarchy. This result is a nontrivial generalization of Borodin's formula for the discrete gap probabilities of the classical random partitions model when the measure is the Poissonized Plancharel measure. The result is obtained by using the Riemann-Hilbert problem (due to Baik-Deift-Johansson) associated to the family of orthogonal polynomials on the unit circle connected to the Toeplitz determinants of interest.

The talk is based on a joint work with Thomas Chouteau (Université d'Angers), soon on the ArXiv!

**Giulio Ruzza ** (UC Louvain) [3:30pm-5pm]

*Cylindrical Toda equation for multiplicative statistics of the discrete Bessel process*

Abstract: The discrete Bessel process is a discrete determinantal point process arising in connection with the distribution of the longest increasing subsequence of a random permutation. I will report on a joint work with Mattia Cafasso (arXiv:2207.01421) where we use a discrete Riemann-Hilbert approach to Fredholm determinants expressing multiplicative statistics of this process (following a general method due to Its-Izergin-Korepin-Slavnov in the continuous case and later adapted by Borodin-Deift to the discrete case). We show by this method that such expectations are tau functions for the cylindrical Toda equation. Further applications include a discrete version of the Amir-Corwin-Quastel integro-differential Painlevé II equation and asymptotic analysis.

## 2021-2022

**UCL/KUL Integrable systems and Classical analysis Seminar (Sept. 2021 – June 2022)**

**Sergey Berezin** (KULeuven), May 10 (2022)

*Products of Ginibre matrices in the critical regime: the gap probability for the soft-edge scaling limit*

Consider a product of i.i.d. Ginibre matrices, and suppose the number of the factors in the product and the sizes of the matrices are asymptotically of the same order (the critical regime). The behavior of the largest singular value is governed by the critical kernel discovered by Dang–Zheng Liu, Dong Wang and Yanhui Wang, not so long ago. We will talk about the gap probability for this kernel, discuss an analog of the Tracy–Widom formula, and present a small gap probability asymptotics. Talk is based on joint work with Eugene Strahov.

**Alexi Morin-Duchesne **(University of Ghent), April 12 (2022)

*Universality and conformal invariance in percolation models*

Abstract: In this talk, I will describe our investigations of the universal behaviour of two critical percolation models: site percolation on the triangular lattice and bond percolation on the square lattice. Both are Yang-Baxter integrable models that can in principle be solved exactly. In the scaling limit, they are conformally invariant and described by non-unitary representations of the Virasoro algebra. I will describe our calculation of the models' partition functions on the cylinder and torus, and how this is related to these Virasoro representations. This is joint work with A. Klümper and P.A. Pearce.

**Johannes Forkel** (Mathematical Institute, University of Oxford), March 22 (2022)

*Moments of Moments of the characteristic polynomial of matrices from the classical compact groups*

This talk is on joint work with Tom Claeys and Jon Keating. Considering the characteristic polynomial $p_n(\theta)$ of a random matrix $G$ as a function on the unit circle, its moments of moments are defined by first taking the $2\alpha$-th moment w.r.t. $\theta$, and then taking the $m$-th moment w.r.t. the underlying $G$. Using results from Claeys, Glesner, Minakov and Yang on uniform asymptotics of Toeplitz+Hankel determinants we established asymptotic formulas for the moments of moments of the characteristic polynomial of random orthogonal and symplectic matrices, as the matrix size $n$ goes to infinity. Those results are analogous to the results from Fahs for the unitary group. As a corollary we obtained results on the range of parameters where certain Selberg-type integrals are finite.

**Samuel Belliard **(IDP, Université de Tours), March 11 (2022)

*Modified Algebraic Bethe Ansatz*

I will discuss a way to calculate spectrum and states for models without U(1) symmetries such as the open XXZ spin chain. The modified algebraic Bethe ansatz will be described in that case, and off-shell action of the related transfer matrix will be established. It will allows to calculate Slavnov’s formula for scalar product of the associated states.

**Augustin Lafay** (LPENS, Paris), February 25 (2022)

*Web models as generalizations of statistical loop models*

Two dimensional gases of non-intersecting loops have been a subject of study in mathematical physics for more than thirty years because of their numerous connections to integrability, two dimensional conformal field theory, random geometry and combinatorics. In this talk, I will present a natural generalization of loop models to gases of graphs possessing branchings. These graphs are called webs and first appeared in the mathematical community as diagrammatic presentations of categories of representations of quantum groups. The web models possess properties similar to the loop models. For instance, it will be shown that they describe, for some tuning of the parameters, interfaces of spin clusters in Zn spin models. Focusing on the numerically more accessible case of Uq(sl3) webs (or Kuperberg webs), it is possible to identify critical phases that are analogous to the dense and dilute phases of the loop models. These phases are then described by a Coulomb Gas with a two component bosonic field.

**Andrea Sportiello **(LIPN, Université Sorbonne Paris Nord), February 1st (2022)

*Many new conjectures on Fully-Packed Loop configurations*

The Razumov--Stroganov conjecture revolves around Fully-Packed Loop configurations (FPL) and the steady state of the Dense O(1) Loop Model (O(1)DLM). In short, the enumeration of FPL's refined according to the (black) link pattern is proportional to the aforementioned steady state. It exists in two main flavours: ‘‘dihedral” (ASM, HTASM, QTASM,... vs. the DLM on the cylinder), and ‘‘vertical” (VSASM, UASM, UUASM, OSASM, OOASM,... vs. the DLM on the strip). Together with L. Cantini, we gave two proofs (in 2010 and 2012) of the conjecture in the dihedral cases, but, despite the efforts of ourselves and others, the vertical case is still unsolved. We recently looked back at the FPL configurations pertinent to one of the unsolved cases, namely the UASM (ASM on a $2n x n$ rectangle with U-turn boundary conditions on one long side), and we had the idea of looking at the refinement according to the black and white link patterns, and the overall number of loops. This doesn't seem to help in understanding the Razumov--Stroganov conjecture, but leads to many more conjectures, suggesting the existence of a remarkable deformation of the Littlewood--Richardson coefficients, somewhat in the same spirit, but apparently by a completely different mechanism, to ‘‘FPL in a triangle’’ studied by P. Zinn-Justin, and by Ph. Nadeau. Work in collaboration with L. Cantini.

**Guilherme Silva** (ICMC, Universidade de São Paulo), December 14 (2021)

*Multiplicative Statistics For Eigenvalues Of Hermitian Matrix Models Are (KPZ) Universal*

We study the large matrix limit of a family of multiplicative statistics for eigenvalues of hermitian matrix models, showing that they universally connected with the integro-differential Painlevé II equation and, in turn, with the KPZ equation. But the connection does not stop there, and we will also explain how the norming constants of the associated orthogonal polynomials and the underlying correlation kernel are asymptotically described in terms of the same solution to the integro-differential PII. Although we work under the assumption of a regular one-cut potential and a family of multiplicative statistics satisfying certain regularity conditions, we also plan to discuss how our approach indicates that other classes of potentials may give rise to different families of integrable systems. Based on ongoing work with Promit Ghosal (MIT).

**Jules Lamers **(IPhT, CEA Saclay), December 7 (2021)

*Recent advances for integrable long-range spin chains*

Long-range interactions are important in cold-atom experiments and possibly for quantum computing. Exactly solvable long-range models provide an arena to test which paradigms for short-range models survive in the presence of long-range interactions. I will introduce the landscape of quantum-integrable long-range spin chains. Unlike for the nearest-neighbour Heisenberg spin chains, the exact solvability/quantum integrability of long-range spin chains is based on connections to integrable quantum-many body systems of Calogero–Sutherland (or Ruijsenaars–Macdonald) type. For the Haldane–Shastry spin chain (and its XXZ-like analogue) this connection yields enhanced Yangian (or quantum-loop) symmetries as well as explicit eigenvectors in terms of Jack (or Macdonald) polynomials. The Inozemtsev spin chain interpolates between Heisenberg and Haldane–Shastry while being exactly solvable throughout. Here the spectrum can be found in terms of eigenfunctions of the elliptic Calogero–Sutherland model, although the underlying quantum-algebraic structure is not understood yet. My talk is based on joint work with R. Klabbers (Nordita), with V. Pasquier and D. Serban (IPhT), and work in progress.

**Sung-Soo Byun** (Korea Institute for Advanced Study), November 16 (2021)

*Lemniscate ensembles with spectral singularity*

In this talk, I will discuss a family of determinantal Coulomb gases, which tend to occupy lemniscate type droplets in the large system. For these lemniscate ensembles under the insertion of a point charge, I will present the scaling limits at the singular boundary point, which are expressed in terms of the solution to the Painlevé IV Riemann-Hilbert problem. The main ingredients of the proof include a version of the Christoffel-Darboux identity and the strong asymptotic behaviour of the associated orthogonal polynomials. This is based on joint work with Seung-Yeop Lee (USF) and Meng Yang (UCPH).

**Eric Ragoucy** (LaPTh, Annecy), September 30th (2021)

*Integrability in out-of-equilibrium systems*

Out-of-equilibrium systems have nowadays an important role in 1d statistical physics. Although an equilibrium state obviously doesn’t exist for such systems, one looks for a steady state (that is stationary in time). It is defined as the zero-eigenvalue eigenstate of the Markov matrix that describe the evolution of the system. Its exact computation is at the core of many researches.

In some cases, the matrix ansatz allows to compute this steady state. However no general approach for this ansatz is known. On the other hand, many 1d statistical models appear to be integrable, which allows to get eigenstates of the Markov matrix through Bethe ansatz. The goal of this presentation is to show how integrability gives a natural framework to construct the matrix ansatz for 1d systems with boundaries. It can be done on very general grounds, allowing to construct the matrix ansatz when it is not known, and also to define new models and/or to find boundary conditions ‘adapted’ to the model under consideration. We will illustrate the technique on some examples.

**Matthias Schötz** (Université Libre de Bruxelles), September 14 (2021)

*Gel'fand Naimark Theorems for ordered *-algebras*

An ordered *-algebra is a (always unital) *-algebra over the field of complex numbers whose Hermitian elements are endowed with a partial order fulfilling some mild compatibilities with the algebraic structure, or equivalently, a *-algebra endowed with a ‘‘quadratic module” of ‘‘positive” Hermitian elements. Basic examples are *-algebras of complex-valued functions with the pointwise operations and the pointwise order, or *-algebras of adjointable endomorphisms of a pre-Hilbert space with the operator order. Another class of examples are (pre-)C*-algebras (i.e. *-algebras which are endowed with a C*-norm), which can always be equipped with a natural order. For pre-C*-algebras, the Gel'fand Naimark theorems show that there always exists a faithful representation as adjointable endomorphisms on a Hilbert space, and that commutative pre-C*-algebras always admit a faithful representation as continuous complex-valued functions on a compact Hausdorff space. The aim of the talk is to discuss in which sense ordered *-algebras can be seen as generalizations of pre-C*-algebras, and to extend the Gelfand Naimark theorems to a more general class of ordered *-algebras, especially to possibly ‘‘unbounded” ones.

**Jean-Marie Stéphan** (Institut Camille Jordan, Lyon) September 1st (2021)

*Super-universal corner contributions to the scaling of fluctuations*

Understanding the fluctuations of observables is one of the main goals in physics, be it theoretical or experimental, quantum or classical. We investigate such fluctuations when only a subregion of the full system can be observed, focusing on geometries with sharp corners. The dependence on the opening angle turns out to be super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. In this talk, I will explain how such questions may be translated into a well-posed asymptotic problem, and how useful physical information may be extracted from the prefactor.

## 2020-2021

Classical analysis and integrable systems online seminar

The seminar was jointly organized between KU Leuven and UC Louvain-la-Neuve.

## Talks 2016

Wednesday, 19 October 2016 from 15:00 to 17:00 at Marc de Hemptinne (CYCL08)

**Manuela Girotti **(UCLouvain)

*"Integrable" gap probabilities for the Generalized Bessel process*

We consider the gap probability for the Generalized Bessel process, a determinantal point process which arises as critical limiting kernel near the hard edge of the spectrum of a certain random matrix ensemble. We prove that such probability can be expressed in terms of the Fredholm determinant of a suitable Its-Izergin-Korepin-Slavnov integrable operator and linked in a canonical way to Riemann-Hilbert problem. Starting from the RH problem, we can construct a Lax pair and we can link the gap probability to the Painleve III hierarchy. Moreover, we are able to construct a system of two coupled Hamiltonians which can be hopefully identified with the 2-dimensional Garnier system LH(2+3).

The talk is based on some previous results and an on-going project with Dr. Mattia Cafasso (Université Angers, France).

Wednesday, 12 October 2016 from 15:00 to 17:00 at Marc de Hemptinne (E349)

**Pieter W. Claeys **:

*Richardson-Gaudin models: What can we learn from (breaking) symmetry ?*

Richardson-Gaudin integrable models hold a special place within the general framework of manybody systems. While their underlying algebraic structure allows for an exact solution by Bethe Ansatz, the large freedom left in their construction allows them to be linked to an extensive range of

physical phenomena such as superconductivity and quantum magnetism.

In this talk, I will focus on the class of hyperbolic Richardson-Gaudin models. After a general introduction to these models, their remarkable symmetries will be introduced and discussed. It will be shown how these symmetries can be exploited in order write down expressions for physical

observables as determinants of matrices. Furthermore, by breaking the u(1) symmetry of these models, the subtle interplay between different symmetry sectors can be uncovered, shedding some light on the structure of the (Bethe Ansatz) wave functions. Throughout this talk, the general theory

will be applied to a specific model which describes topological superconductivity, highlighting the physical consequences of these results.

The talk will mainly be based on these two recent papers :

[1] P. W. Claeys, S. De Baerdemacker, M. Van Raemdonck, and D. Van Neck, Phys. Rev. B 91, 155102 (2015).

[2] P. W. Claeys, S. De Baerdemacker, and D. Van Neck, Phys. Rev. B 93, 220503(R) (2016).

Wednesday, 5 October 2016 from 15:00 to 17:00 at Marc de Hemptinne (CYCL08)

**Yacine Ikhlef**, LPTHE (Univ. Paris-6 and CNRS)

*"Operator algebra in critical loop models and non-rational Conformal Field Theories"*

Abstract:

Loop models are lattice statistical models with non-local Boltzmann weights, which generally describe extended geometrical objects such as spin interfaces in the Ising model, or percolation clusters. Since the late 80s, they have been recognised as lattice realisations of non-rational Conformal Field Theories (CFTs), with a discrete but infinite spectrum of scaling dimensions. However, the operator algebra of these models is not accessible by the standard methods of CFT. Recently, this question regained interest when Delfino and Viti (2010) showed that the structure constant from the "time-like Liouville" theory gives the correct value for the three-point connectivity of percolation clusters. I will present some related results on the O(n) loop model, including the extension of the Delfino-Viti approach to a large class of scalar operators and the bootstrap analysis for non-scalar operators in CFTs based on the Virasoro and W_3 algebras.

Wednesday, 1 June 2016, 11:00 to 12:30 [CYCL08]

**Grégory Schehr** (Orsay)

*"Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time"*

Abstract:

I will consider a system of N one-dimensional free fermions confined by a harmonic well. At zero temperature (T=0), it is well known that this system is intimately connected to random matrices belonging to the Gaussian Unitary Ensemble. In particular, the density of fermions has, for large N, a finite support and it is given by the Wigner semi-circular law. Besides, close to the edges of the support, the quantum fluctuations are described by the so-called Airy-Kernel (which plays an important role in random matrix theory). What happens at finite temperature T ? I will show that at finite but low temperature, the fluctuations close to the edge, are described by a generalization of the Airy kernel, which depends continuously on temperature. Remarkably, exactly the same kernel arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time. I will also discuss recent results for fermions in higher dimensions

28/4/16, 16h15 [CYCL08]

**Christian Hagendorf** (UCL)

*"An introduction to integrable vertex models (3)"*

21/4/16, 16h15 [CYCL08]

**Christian Hagendorf** (UCL)

*"An introduction to integrable vertex models (2)"*

14/4/16, 14h00 [CYCL08]

**Christian Hagendorf** (UCL)

*"An introduction to integrable vertex models (1)"*

18/2/16, 16h15 [CYCL08]

**Christophe Charlier and Tom Claeys** (UCLouvain)

*"Thinning and conditioning of spectra of random unitary matrices"*

4/2/16, 16h15 [CYCL08]

**Raoul Santachiara** (Orsay)

"Conformal invariance and many-point correlation functions in percolation models"

Abstract: The geometrical nature of two-dimesional critical points is encoded in the connectivity properties of random clusters. We discuss recent results on the bulk three-point connectivities which were at the origin of our extension of the Liouville theory to values of the central charge $c\leq 1$. Moreover, we present an analytical and numerical analysis of the four-point connectivities of random cluster and pure percolation theory.