The GPP Seminar is the UCL-KULeuven seminar in the framework of the Interuniversity Attraction Pole (IAP) **"Dynamics, Geometry and Statistical Physics" (DYGEST)**. It is alternately held in the mathematics-physics building at UCL (Chemin du Cyclotron 2, Louvain-la-Neuve) or in the mathematics building (Celestijnenlaan 200B, Heverlee) at the KULeuven.

**Organizers** : Alexi Morin-Duchesne (UCL).

# 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.

# Talks 2015

Tuesday 27/10, 11h [CYCL08]

**Jorgen Rasmussen **(University of Queensland)

*"Integrability and conformal data of the dimer model"*

Abstract:

The dimer model concerns the number of ways the edges of a finite graph can be covered by dimers so that each vertex is covered exactly once. We will consider the model defined on the square lattice where the counting exercise amounts to a domino tiling problem. Following Lieb, we will describe the model using a so-called transfer matrix acting on spin chains. Somewhat surprisingly, this reveals an underlying Temperley-Lieb algebraic structure, allowing us to address the integrability and conformal properties of the model. Efforts will be made to keep the presentation accessible to non-experts.

Thursday 22/10, 16h15 [CYCL09B]

**Arno Kuijlaars **(KULeuven)

*"Critical eigenvalue densities in sums of random matrices" *

Thursday 15/10/2015, 16:15 [CYCL08]

**Max Atkin** (UCLouvain)

*"On the separation of the smallest eigenvalues in the Laguerre unitary ensemble"*

Tuesday June 16th at 11:00

**Nicolas Allegra** (U. Nancy)

*"Dimer model: What can possibly be new ?"*

Abstract : In the first part of this presentation, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented and some important applications will be detailled and compared to CFT results via the so-called height mapping. In a second part a one-dimensional model of free fermionic particles evolving in imaginary time from a domain-wall initial state (DWIS) is introduced and solved. The main interest of this model is that it exhibits the ''arctic-circle phenomenon'' originally discovered in dimer models by Propp, Jokusch, Shor. Large-scale correlations inside the fluctuating region are expressed in terms of correlators in a (euclidean) massless Dirac theory in curved two-dimensional space.

CYCL08, Marc de Hemptinne, chemin du Cyclotron 2, 1348 Louvain-la-Neuve

Wednesday April 1, 2015, 16h15, Cycl08

**Igor Krasovsky** (Imperial College London)

*"Asymptotic behaviour of a sine-kernel determinant which appears in the theory of random matrices and log-gases"*

Thursday 26/2/15, 16h15, CYCL08

**Ward Struyve** (Université de Liège)

*"Bohmian mechanics"*

Abstract: Bohmian mechanics is an alternative to standard quantum mechanics that describes point-particles that move under the influence of the wave function. The main motivation to consider such an alternative is that it solves the conceptual problems, such as the measurement problem, that plague quantum mechanics. I will give an introduction to the theory, review its current status, and touch upon some recent applications in cosmology.

Wednesday 04/02, 14h, CYCL07

**Arno Kuijlaars** and **Dries Stivigny** (KULeuven)

*"Products of truncated unitary matrices"*

# Talks 2014

Thursday November 27, 16h15 , CYCL09b,

**Guilherme Silva** (KULeuven)

"Breaking the symmetry in the normal matrix model"

Abstract. We consider the normal matrix model with cubic + linear potential. The model is ill-defi ned, and to regularize it, Elbau and Felder proposed to make a cut-o ff on the complex plane, leading to a system of orthogonal polynomials with respect to a certain 2D measure. When studying this model with a monic cubic weight, Bleher and Kuijlaars associated to it a system of non-hermitian multiple orthogonal polynomials, which are expected to be asymptotically the same as the 2D orthogonal polynomials. In this talk, the focus will be on the non-hermitian MOP's in the spirit of Bleher and Kuijlaars, but now adding a linear term to the cubic potential. It will be shown how some quantities of the normal matrix model are related to those orthogonal polynomials. At the technical level, the linear term breaks the symmetry of the model, and in order to deal with it a quadratic di fferential on the spectral curve is introduced, and its critical trajectories are described globally. These trajectories play a fundamental role on the steepest descent analysis of the associated Riemann-Hilbert problem, as will be explained if time permits.

This is an ongoing project with Pavel Bleher (Indiana University - Purdue University Indianapolis).

Thursday 30/10, 14h00, CYCL02

**Mattia Cafasso** (Université d'Angers)

"Block Toeplitz determinants and tau functions for Gelfan'd Dickey hierarchies"

Abstract: In this talk I will explain how the Szegö-Widom formula for the asymptotics of block Toeplitz determinants can be used to identify certain large Toeplitz determinants with tau functions for Gelfan'd Dickey hierarchies. Connections with Riemann-Hilbert problems and isomonodromic tau functions will be discussed, as well as some explicit computations (time permitting).

Thursday 16/10, 14h, CYCL02

**Roman Riser** (KULeuven)

Universality of Higher Order Corrections in Random Normal Matrices

Abstract: We will conjecture a universal formula for the corrections to the averaged density of eigenvalues for n x n random normal matrices up to order O(1/n). At the boundary of the support of the density it only depends on local geometrical quantities like the curvature of the boundary.

Wednesday, June 4, 16:15, CYCL b328, UCL Math building, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve

**Benjamin Niedner** (University of Oxford)

"New Boundary Conditions for the Potts Model on Random Planar Maps"

Abstract: We revisit the 3-states Potts model on random planar triangulations as a Hermitian matrix model. As a novelty, we obtain an algebraic curve which encodes the partition function of the model on the disc with both fixed and mixed spin boundary conditions. We investigate the critical behaviour of the model, find scaling exponents consistent with previous literature and argue that the highest critical point is described by a rational conformal field theory with W-symmetry coupled to Liouville gravity.

Thursday May 15, 2014 [Louvain-La-Neuve, bâtiment de Hemptinne, CYCL09]

**DYGEST workshop WP2 **

Dynamical and integrable systems

14h00 **Tamara Grava** (SISSA Trieste, University of Bristol) - Asymptotic expansion of Hermitian matrix integrals

15h00 **Thorsten Neuschel** (KULeuven) - Asymptotics for average characteristic polynomials of Wishart type products of random matrices

16h00 Coffee break

16h15 **Kurt Johansson** (KTH Stockholm) - The two-periodic Aztec diamond

Wednesday 26/03/2014, 16h15, b328

**Alexi Morin-Duchesne** (UCL)

"Functional relations in Temperley-Lieb loop models"

Abstract: Planar algebras have found applications in the description of lattice models in statistical mechanics with non-local degrees of freedom. Of particular importance are the diagrammatic objects known as transfer tangles since certain representations thereof correspond to lattice model transfer matrices. Working in the planar Temperley-Lieb algebra, I will discuss how the corresponding transfer tangles satisfy various functional relations. A fusion procedure based on applications of the Wenzl-Jones projectors is used to describe and prove these relations. This approach gives rise to a fusion hierarchy which can be reformulated as so-called Y-systems of importance in the analysis of integrable models.

Wednesday 12/03/2014, 16h15, b328

**Dries Stivigny** (KULeuven)

" Products of random matrices "

Abstract: Recently, Akemann et al. showed that the squared singular values of products of complex Ginibre random matrices give rise to a determinantal point process. We show more generally that the product with a complex Ginibre random matrix preserves the determinantal structure. Using this theorem we obtain the joint probability density function for the squared singular values of the product of a truncated unitary matrix with M-1 complex Ginibre random matrices. For this ensemble, we look at the correlation kernel and obtain an integral representation. Using this, we calculate the limiting kernel at the origin (hard edge). We obtain the same kernel as Kuijlaars and Zhang obtained for the case of the product of M complex Ginibre random matrices. We end this talk by shortly discussing an example, not coming from the product of random matrices, where the same kernel also appears. This is joint work with Arno Kuijlaars.

Wednesday February 19, 16:15 [Bâtiment de Hemptinne, b328]

**Manuela Girotti** (Concordia University, Montréal, Canada)

"Gap Probabilities of the Tacnode process"

Abstract: We study the gap probabilities of the Tacnode process, expressing them as the ratio of two explicit Fredholm determinants. The formula allows us to build a Riemann-Hilbert problem and, through suitable steepest descent analysis, we will be able to show that, under appropriate scaling regimes, the gap probability of the Tacnode process degenerates into a product of two independent gap probabilities of the Airy processes.

Wednesday January 29, 16:00 [Bâtiment de Hemptinne, b328]

**Marta Mazzocco** (Loughborough University)

"Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme"

Abstract: In this talk we establish a link between q–orthogonal polynomials and the theory of Painlevé equations. The Painlevé differential equations were discovered more than a hundred years ago and since then have been subject to intense study by many mathematicians worldwide. The Painlevé differential equations possess many beautiful properties, for example they are ”integrable" in the sense that they can be written as compatibility conditions between two linear systems of the first order. This compatibility allows us to establish the so-called Riemann Hilbert correspondence which associates to each local solution of a given Painlevé equation a point on a cubic surface called monodromy manifold. Basic hypergeometric polynomials are (usually Laurent) polynomials which can be expressed in terms of the q-Hypergeometric series. The Askey Wilson polynomials are the most general one variable basic hypergeometric polynomials. They span the eigenspaces of the basic representation of the Askey Wilson algebra (a subalgebra of certain Cherednik algebra). We prove that this algebra is the quantisation of the monodromy manifold associated to the sixth Painlevé equation. By confluencing the Painlevé equations, we construct other new algebras with the property that their basic representation is spanned by confluences of the Askey Wilson polynomials belonging to the q-Askey scheme. This answers the open problem of associating a quantum algebra to each element of the q-Askey scheme. Incidentally, in this construction the orthogonal polynomials appear for the first time on the right hand side of the Riemann-Hilbert correspondence, rather than on the left as in all previous publications on this subject.

# Talks 2013

Wednesday December 4, 16:15 [Bâtiment de Hemptinne, b328]

**Vladimir Mitev** (HU Berlin)

"Dynamical Lattice Supersymmetry in GL(m|n) Spin Chains"

Abstract: We attempt to classify, via an analysis of the Bethe equations, all homogeneous integrable spin chains for which we expect length-changing supersymmetry generators to be present. We are able to write down the supercharges explicitly for the simplest models, namely the sl(n|1) spin chains with the (n-1)-fold antisymmetric tensor product of the fundamental representation at each site.

Wednesday November 27, 16:15 [Bâtiment de Hemptinne, b328]

**Kalle Kytola** (University of Helsinki)

"SLE boundary zig-zags and multiple SLEs by quantum groups "

Abstract: Schramm-Loewner evolutions (SLE) are conformally invariant random curves that describe scaling limits of interfaces in various models of critical statistical physics in two dimensions. In this talk we consider two questions related to SLEs. The first question is about the "boundary zig-zags", i.e. the probabilities for a chordal SLE to pass through small neighborhoods of given boundary points in a given order. The second question is that of obtaining explicit descriptions of "multiple SLE pure geometries", i.e. those extremal multiple SLE probability measures which can not be expressed as non-trivial convex combinations of other multiple SLEs. For both problems one needs to find solutions of a system of partial differential equations with asymptotics conditions written recursively in terms of solution of the same problem with a smaller number of variables. We present a general correspondence, which translates these problems to linear systems of equations in finite dimensional representations of the quantum group U_q(sl_2), and we then explicitly solve these systems. The talk is based on joint works with Eveliina Peltola (Helsinki), and with Niko Jokela (Santiago de Compostela) and Matti Järvinen (Crete) and some ongoing work with Konstantin Izyurov (Helsinki).

Wednesday October 23, 17:00 [Bâtiment de Hemptinne, b328]

**Guilherme L. F. Silva** (KULeuven)

*"S-curves and (non-hermitian) orthogonal polynomials"*

__Abstract__

Thursday October 17, 14:00-16:00 [KU Leuven, Celestijnenlaan 200B, Room 02.18]

**Christophe Charlier** (UCL)

"Asymptotic study for certain Toeplitz determinants "

Thursday October 10, 16h30 [Bâtiment de Hemptinne, CYCL01]

** Kenneth D. T.-R. McLaughlin **(University of Arizona)

*"Random matrices: an overview, and connection to the asymptotic behavior of the zeros of the Taylor approximants of the exponential function"*

Abstract: I will start with an introduction to several random matrix ensembles and discuss asymptotic properties of the eigenvalues of the matrices. Following that, I will make the connection to the asymptotic behavior of the zeros of the Taylor approximants of the exponential function.

(Colloquium talk)

Thursday 23/5, 16:15 [CYCL b328]

**Max Atkin** (Bielefeld)

*From Instantons to Large Deviations in Hermitian Random Matrices*

Abstract: The typical fluctuations of the largest eigenvalue of a hermitian random matrix about its mean

has been known for a long time to be given by the Tracy-Widom distribution. More recently interest has

focused on the 'atypical' fluctuations in which an eigenvalue appears very far from the bulk spectrum.

This problem has been attacked using saddle point methods, loops equations and to a much lesser extent,

orthogonal polynomials. In this talk we review recent progress in the orthogonal polynomial approach

which makes contact with instanton effects in the string theory literature. We use this framework to derive

the distribution for large deviations in the case of multi-critical potentials.

Thursday 25/4, 16:15 [CYCL b328]

**Neil O'Connell **(University of Warwick)

*Geometric RSK, Whittaker functions and random polymers: raising the temperature in random matrix theory *

Wednesday, March 6, 2013 [CYCL02]

**DYGEST Workshop on Integrable Systems**

**Hendrik De Bie** (Ghent University)

Closed expressions for kernels of generalized Fourier transforms

**Stefano Romano** (UCLouvain)

Two paths from integrable systems to Frobenius Manifolds

**Lun Zhang** (KULeuven)

Universality and critical behavior in the chiral two-matrix model

**Arno Kuijlaars** (KULeuven)

The tacnode Riemann-Hilbert problem

Thursday 21/2, 16:15 [CYCL b328]

**Andrea Raimondo**

SISSA, Trieste

*"A deformation of the method of characteristics and the Cauchy problem for Hamiltonian PDEs in the small dispersion limit"*

Abstract;: In this talk, I will consider a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. After providing an explicit construction up to the fourth order in perturbation theory, I will rigorously prove the validity of the new equation in the small dispersion limit. I will then apply the deformed characteristics method to explicitly compute the first two perturbative corrections of the solution of the general Hamiltonian PDE. In the case of the KdV equation, our construction gives an affirmative answer to the conjecture of quasi-triviality proposed by Dubrovin [Comm. Math. Phys. , 267(1), 2006].

Friday 1/2/13, 11:00 [CYCL b328]

**Gernot Akemann **(Bielefeld University)

*"Spectral correlations of products of random matrices from Ginibre ensembles "*

# Talks 2012

Thursday 17/1/13, 16:15 [CYCL b328]

**Sunil Chhita**

KTH (Stockholm)

*"Coupling functions of domino tilings of Aztec Diamonds" *

Abstract: The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the graph such as its local statistics. These statistics can be used to find global and local asymptotic behavior.

In this talk, we present three different weightings (one-periodic, two-periodic and q^{vol} weightings) of the Aztec diamond each with different but striking limiting behavior. We show that it is possible to find recurrence relations which allows a derivation of the inverse Kasteleyn matrix. This is joint work with Benjamin Young (University of Oregon).

Thursday 6/12, 16h15 [CYCL b328]

**Raphael Lefevere**

Univ Paris Diderot, France

*Macroscopic diffusion from a Hamilton-like dynamics*

Abstract: We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model has all the properties of Hamiltonian dynamics in a confined phase space : it is deterministic, periodic, reversible and conservative. Randomness enters the model as a way to model ignorance about initial conditions and interactions between the components of the system. The orbits of the particles are non-intersecting random loops. We prove, as a weak law of large number, the validity of a diffusion equation for the macroscopic observables of interest for time arbitrary large, but small compared to the minimal recurrence time in the dynamics.

Tuesday 4/12, 16h15 [CYCL b328]

**Alexander Tovbis**

University of Central Florida, US

*Painlevé transcendents and universality of transitions at the point of gradient catastrophe for semiclassical (small dispersion) limit of the Nonlinear Schrödinger (NLS) equation: the Riemann-Hilbert Problem approach.*

Abstract: The semiclassical (zero-dispersion) limit of solutions $q=q(x,t,\e)$ to the one-dimensional focusing Nonlinear Schröodinger equation (NLS) is studied in a scaling neighborhood $D$ of a point of gradient catastrophe ($x_0,t_0$). This neighborhood contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast amplitude oscillations (spikes). We establish the following universal behaviors of the NLS solutions $q$ near the point of gradient catastrophe:

i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size $O(\ve)$;

ii) the location of the spikes is determined by the poles of the tritronqu\'ee solution of the Painlevé I (P1) equation through an explicit map between $D$ and a region of the Painlevé independent variable;

iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t,\e)$ is given by the plane wave approximation $q_0(x,t,\e)$, with the correction term being expressed in terms of the tritronqu\'ee solution of P1.

The relation with the conjecture of Dubrovin, Grava and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed.

Thursday 29/11, 16h15 [CYCL b328]

**J. P. Gazeau APC,**

Univ Paris Diderot, France

*Distributions binomiales généralisées*

Abstract: In most of the realistic models in Physics one must take correlations into account; events, which are usually presented as independent, like in a binomial Bernoulli process, are actually submitted to correlative perturbations. These perturbations lead to deformations of the mathematical independent laws. In accordance with this statement, we present a generalization of the binomial distribution associated with a sequence of positive numbers. It involves asymmetric and symmetric expressions of probabilities for (win-loss) sequences of trials. Our approach is based on generating functions and produces, in the symmetric case, polynomials of the binomial type. Poisson-like limits, Leibniz triangle rules and related entropy(ies) are considered. Our generalizations are illustrated by various analytical and numerical examples.

Thursday, November 15, 2012 [KULeuven - Celestijnenlaan 200B, Room 02.18]

### DYGEST Workshop on Integrable Systems

**Maarten van Pruijssen** (Radboud Universiteit Nijmegen, Netherlands)

*"Matrix valued classical pairs from group theory"*

**Pablo Manuel Román** (Universidad Nacional de Córdoba, Argentina)

*"Explicit expressions for matrix valued orthogonal polynomials related to spherical functions of any K-type"*

**Tom Claeys **(Université Catholique de Louvain, Belgium)

*"The partition function in two-cut random matrix ensembles"*

Thursday 22/3 - 16:15 [room FYMA -CYCL b 328, UCL]

**Alfredo Deaño** (Universidad Carlos III de Madrid)

*"Partition function and free energy in the cubic random matrix model"*

Abstract | We consider a unitary random matrix model with weight function e^{-NV(z)}, where V(z)=z^2/2-uz^3 and u is a real positive parameter. One feature of this model is that the partition function has to be defined in the complex plane. For u smaller than a critical value u_c, the free energy admits an asymptotic expansion in powers of N^{-2}. The first two terms of this topological expansion are known from the work of Brézin, Itzykson, Parisi, and Zuber (Commun. Math. Phys. 59 (1978), 35-51), and can be written in terms of classical hypergeometric functions. Near the critical value, a double scaling limit leads to an asymptotic approximation in terms of a particular solution of Painlevé I. Our results are obtained by a Riemann--Hilbert analysis of the corresponding family of orthogonal polynomials in the complex plane, together with the string equations for the associated recurrence coefficients. (joint work with Pavel M. Bleher, Indiana University-Purdue University Indianapolis, USA)

Friday 9/3 - 16h15 [room FYMA -CYCL b 328, UCL]

**Stéphane Korvers** (UCL)

*"Étude des aspects géométriques et analytiques d’une extension d’un processus de quantification par déformation de l’espace symétrique symplectique SU(1,1)/U(1)"*

Abstract | Dans « The Deformation Quantizations of the Hyperbolic Plane » (P. Bieliavsky, S. Detournay, Ph. Spindel ; Communications in MathematicalPhysics ; Springer-Verlag 2008) les auteurs montrent qu'une contraction de courbure du plan hyperbolique livre une surface symétrique symplectique dont le groupe de transvections est isomorphe au groupe de Poincaré en dimension deux. Ils montrent également que de ce processus de contraction émerge un opérateur différentiel d'ordre deux dont certaines solutions de l'évolution définissent des opérateurs de convolution qui entrelacent la théorie des déformations (produits-étoile) au niveau contracté avec celle du plan hyperbolique. Cet exposé consistera en un bref aperçu de la généralisation de cette construction au cas de l'espace symétrique symplectique SU(1,n)/U(n) et en l’analyse des résultats annexes obtenus à partir d’une étude géométrique de l’opérateur différentiel d'ordre deux dans le cas du plan hyperbolique. [±]

Friday 24/2 - 16h15 [room FYMA -CYCL b 328, UCL]

**Mark Adler** (Brandeis University)

*"RMT minors and percolation Theory "*

Thursday 9/02 (16:15) [room FYMA -CYCL b 328, UCL]

**Daniela Rosca** (Univ. Techn. Cluj, Roumania)

*"Area preserving maps, uniform grids, and the associated wavelet analysis"*

Abstract | We give three different methods for constructing some area preserving bijections. Then we apply these bijections to uniform rectangular grids and we obtain uniform grids on the disc, elliptic domains and surfaces of revolution. Combined with the Lambert azimuthal projection, our new maps can be used to obtain new uniform and refinable (UR) grids on the sphere, starting from (UR) grids on rectangles. At the end we show how a given wavelet analysis of R^2 can be transported to surfaces of revolution.

Thursday 26/01 (16:15) [ room FYMA -CYCL b 328, UCL]

**Jean-Pierre Antoine** (UCL)

*"Partial inner product spaces, a unifying concept in functional analysis (2)"*

Abstract | Many families of function spaces play a central role in analysis, in particular in signal processing (e.g. wavelet or Gabor analysis). Such are Lp spaces, Besov spaces, amalgam spaces or modulation spaces. In all such cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. In this context, it is often said that such families should be taken as a whole and operators, bases, frames on them should be defined globally, for the whole family, instead of individual spaces. It turns out that all these space families are scales or lattices of Banach spaces, and as such they are special cases of partial inner product spaces (pip-spaces). These objects, which may often be seen as an alternative to the theory of tempered distributions, have been studied systematically in many papers and are now the subject of a recent monograph [1]. The interesting fact is precisely that they allow a global definition of operators, and various operator classes on them have been defined. In these talks, we shall give an overview of pip-spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. [1] J-P. Antoine and C. Trapani, Partial Inner Product Spaces — Theory and Applications; Lecture Notes in Mathematics, vol. 1986, Springer-Verlag, Berlin, Heidelberg, 2009... [±]

# Talks 2011

Thursday 22/12 (16h 15) [room FYMA - CYCL b328, UCL]

**Pierre van Moerbeke** (UCL)

*"The tacnode process (2)"*

Thursday 15/12 (16h 15) [room FYMA - CYCL b 328, UCL]

**Jean-Pierre Antoine **(UCL)

*"Partial inner product spaces, a unifying concept in functional analysis (2)"*

Thursday 08/12 (**14h30**) [room FYMA - CYCL b 328, UCL]

**Jean-Pierre Antoine **(UCL)

*"Partial inner product spaces, a unifying concept in functional analysis (1)"*

Abstract: Many families of function spaces play a central role in analysis, in particular in signal processing (e.g. wavelet or Gabor analysis). Such are Lp spaces, Besov spaces, amalgam spaces or modulation spaces. In all such cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. In this context, it is often said that such families should be taken as a whole and operators, bases, frames on them should be defined globally, for the whole family, instead of individual spaces. It turns out that all these space families are scales or lattices of Banach spaces, and as such they are special cases of partial inner product spaces (pip-spaces). These objects, which may often be seen as an alternative to the theory of tempered distributions, have been studied systematically in many papers and are now the subject of a recent monograph [1]. The interesting fact is precisely that they allow a global definition of operators, and various operator classes on them have been defined. In these talks, we shall give an overview of pip-spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis.

[1] J-P. Antoine and C. Trapani, Partial Inner Product Spaces — Theory and Applications; Lecture Notes in Mathematics, vol. 1986, Springer-Verlag, Berlin, Heidelberg, 2009

Thursday 01/12 (16h 15) [room FYMA - CYCL b 328, UCL]

**Pierre van Moerbeke** (UCL)

*"The tacnode process (1)"*

Thursday 24/11 (16h 15) [room FYMA - CYCL b 328, UCL]

**Tom Claeys** (UCL)

*"Random matrix models with equi-distant external source"*

Abstract: I will discuss random matrix ensembles with a full rank external source. The eigenvalues of the external source are equi-spaced on an interval. I will set up a Riemann-Hilbert problem for the associated multiple orthogonal polynomials and explain how an asymptotic analysis for this problem can be performed. The limiting mean eigenvalue distribution of the model will be described in terms of an equilibrium problem, and bulk and edge universality will be discussed. This is based on joint work in progress with Dong Wang.

Thursday 10/11 (15h 30) [room FYMA - CYCL b 328, UCL]

**Jean-Pierre Gazeau** (Paris 7)

*"Coherent states and related quantizations for unbounded motions"*

Abstract: We present two different methods of construction of coherent states for unbounded motions. The first one adapts the Malkin-Manko approach to such systems and is illustrated by the example of a charged particle in an electric field. The second approach associates to a conservative mechanical system with observed continuous energy spectrum families of corresponding ``pseudo-action-angle'' coherent states. These states are constructed in view to provide a quantization scheme consistent with this continuous range of experimental energies. The formalism is a natural extension of the Bohr-Sommerfeld rule and an alternative to the canonical quantization. Some semi-classical behaviors of such states will be shown.

Thursday 30/6 (16:00) [romm b328]

**Galina Filipuk** (University of Warsaw)

*"Orthogonal polynomials and Painleve equations: Charlier, Meixner and Laguerre cases Location"*

Thursday 12/5 (16h15) [b328]

**Jean-Pierre Antoine**

*"Frames and semi-frames"*

Thursday 5/5 (16h15) [b328]

**Georg Rieckh** (Vienna)

*"Frame Representation of Boundary Integral Operators"*

Thursday 28/4 (16h15) [b328]

**Abey Lopez Garcia **(K.U.Leuven)

*"Some discrete minimal energy problems: asymptotics of Leja and Fekete points"*

Friday 08/04 (16h 15) [room FYMA - CYCL b328, UCL]

**Igor Krasovsky **(Brunel University London)

*"Asymptotics of Toeplitz determinants"*

Thursday 07/04 (15h00) [Fyma b328, UCL]

**Brian Rider** (University of Colorado)

*"Some solvable two-charge models"*

Thursday 31/03 (15h20) [Room B02.18, Celestijnenlaan 200B, KULeuven]

**Dong Wang **(UCL, on leave from University of Michigan)

*"Random matrix models with spiked external source"*

Abstract: In this talk I will show results on random matrix models with spiked (i.e., finite rank) external source. We discuss orthogonal, symplectic and unitary ensembles, and for orthogonal and symplectic ensembles we consider mostly the rank 1 case. Part of the talk is based on joint work with Jinho Baik.

Thursday 10/3 (16h) [FyMa, b328]

**Lun Zhang** (K.U.Leuven)

*"Large n limit of non-intersecting squared Bessel pahts "*

Friday 4/3 (16h) [FyMa, b328]

**Lise Ponselet **(UCL)

*"Transitions de phase des automates cellulaires probabilistes"*

Thursday 24/2 (16h) [FyMa, b328]

**Jean Pierre Gazeau** (Paris 7)

*"Les quantifications canonique et par états cohérents sont-elles physiquement équivalentes?"*

Resumé: Il a été récemment soutenu (Kastrup, 2006) que la quantification canonique est la seule permettant d'expliquer l'observation, remontant aux premières années de la mécanique quantique, de déplacements des raies dans les spectres de vibrations pour des molécules diatomiques isotopiques. Je montrerai que la quantification par états cohérents est une alternative parfaitement valide quant à ses prédictions sur les spectres en question.

# Talks 2010

Thursday 16/12/10 (14:30) [UCL, room FYMA - CYCL b328]

**Jean Bricmont **(UCL)

*"The Fourier law (2)"*

Abstract (Second of two talks)

In this second talk, I will describe some specific microscopic models where one tries to derive Fourier's law, mention some results that have been obtained, and explain the role that "chaotic" dynamical systems might play.

Thursday 09/12/10 (14:30) [UCL, room FYMA - CYCL b328]

**Jean Bricmont **(UCL)

*"The Fourier law (1)"*

Abstract (First of two talks)

Fourier's law on heat conduction is one of the most basic and simplest of the macroscopic physical laws. One of the goals of statistical mechanics is to derive such laws from the microscopic mechanical ones, but this has turned out to be extremely hard.

In the first talk, I will explain the general framework in which such a derivation might be done, and why, although Fourier's law leads to irreversible behaviour, it is not a priori impossible to derive it from reversible microscopic ones; nevertheless, because of this irreversible behaviour, this derivation can only be done, if at all, under certain conditions that will be explained.

Thursday 02/12/10 (14:30) [UCL, room FYMA - CYCL b328]

**Philippe Ruelle **(UCL)

*"From statistical models to conformal field theories *(3)"

Thursday 25/11/10 (14:30) [UCL, room FYMA - CYCL b328]

**Philippe Ruelle **(UCL)

*"From statistical models to conformal field theories *(2)"

Thursday 18/11/10 (14:30) [UCL, room FYMA - CYCL b328]

**Philippe Ruelle** (UCL)

*"From statistical models to conformal field theories (1)"*

Abstract (3 talks, the first one will be an introductory talk)

For the last 25 years, conformal theory has arguably been among the most productive factory of exact -but non rigorous- results in physics and mathematics. It was originally conceived to describe the critical properties of statistical models (f.i. the Ising model), and has indeed led to spectacular progress in the understanding of critical universality classes, especially in two dimensions. The series of talks is meant to be an introduction to the ideas underlying the description of critical models by conformal theories. I will try to give a feeling for the methodology of conformal theories and for some of their technical aspects.

Thursday 10/11/10 (14:30) [UCL, room FYMA - CYCL b328]

**Tom Claeys** (UCL)

*"Asymptotics for the Korteweg-de Vries equation and perturbations using Riemann-Hilbert method (3)"*

Thursday 28/10/10 (14:30) [UCL, room FYMA - CYCL b328]

**Tom Claeys **(UCL)

*"Asymptotics for the Korteweg-de Vries equation and perturbations using Riemann-Hilbert method (2)"*

Thursday 14/10/10 (14:30) [UCL, room FYMA - CYCL b328]

**Tom Claeys** (UCL)

*"Asymptotics for the Korteweg-de Vries equation and perturbations using Riemann-Hilbert method (1)"*

Abstract (3 talks, the first one will be an introductory talk)

Small dispersion asymptotics for solutions to the Korteweg-de Vries equation can be obtained using Riemann-Hilbert problems. In critical regimes, this leads to asymptotic expansions in terms of Painlevé transcendents. I will give an overview of this procedure, and I will discuss some obstacles that occur when considering non-integrable perturbations of the Korteweg-de Vries equation. This is based on joint work with Tamara Grava, and on joint work in progress with Tamara Grava and Ken McLaughlin.

**"IAP Workshop: Integrable Systems 2"**

Wednesday February 10, 10:00 - 16:30

Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, room CYCL 01.

- 10:00 - 11:00

**Pablo Manuel Roman**(KULeuven)

*Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths* - 11:15 - 11:45

**Didier Vanderstichelen**(UCLouvain)

*A centerless representation of the Virasoro algebra associated with the unitary circular ensemble* - 12:00 - 12:30

**Gilles Regniers**(UGent)

*On Wigner quantization of H=xp, representations of osp(1|2) and orthogonal polynomials* - 14:15 - 15:15

**Tom Claeys**(Université Lille 1)

*Critical asymptotics for Toeplitz determinants* - 15:30 - 16:30

**Eric Nordenstam**(UCLouvain)

*Dynamics on Interlaced Particles*

# Talks 2009

Wednesday, 04/11/09 (13:30) [KULeuven, Room B.02.18, Celestijnenlaan 200B, Leuven]

**Eric NORDENSTAM ** (UCLouvain)

*"Determinantal point processes from the perspective of Alexei Borodin"*

**"Workshop:** **Integrable Systems"**

Wednesday May 27 14.30-17.10, Celestijnenlaan 200B, Leuven, room B.00.16.

- 14:30-15:10

**Walter Van Assche**:

*Introduction to multiple orthogonal polynomials* - 15:30-16:10

**Gilles Regniers and Joris Van der Jeugt**:

*Analytically solvable quantum Hamiltonians and relations to orthogonal polynomials* - 16:30-17:10

**Mattia Cafasso**:

*Toda-type equations extended to the case of mixed multiple orthogonal polynomials*

Thursday, 30/04/09 (16:30) [UCL, room FYMA - CYCL b328]

**Marco Bertola **(Concordia University, Montreal)

*"Riemann--Hilbert problems: Malgrange, Schlesinger and Sato"*

**Abstract :** Given any (sufficiently well-behaved) family of Remann--Hilbert problems where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form $\Omega$ on the deformation space (Malgrange's differential). Such a one--form has a pole where the deformation family meets the Malgrange Theta divisor, namely, the set of unsolvable RHP. The differential $\Omega$ fails to be closed in general, but when it does the formula $\tau := {\rm e}^{\int \Omega}$ defines locally a function that vanishes precisely on $\Theta$.

Friday 24/04/09 (11:00) [UCL, room FYMA - CYCL b328]

**Raphaël LEFEVERE** (Université Denis Diderot, Paris 7)

*"Hot scatterers and tracers for the transfer of heat in collisional dynamics"*

**Abstract :** We consider two models for the transport heat in systems described by local collisional dynamics. The dynamics of those models consist of tracer particles moving through an array of hot scatterers describing the effect of heat baths at fixed temperatures. Those models have the structure of Markov renewal processes whose large deviations and ergodic properties are studied in details. When the set of temperatures is fixed by the condition that in average, no energy is exchanged between the scatterers and the system, two behaviours may occur. When the tracer particles are allowed to travel freely through the whole array of scatterers, the temperature profile is linear. If the particles are locked in between scatterers, the temperature profile becomes nonlinear. In both cases, Fourier law holds. Joint work with L. Zambotti.... [±]

Location | Fyma seminar room, b328, Louvain-la-Neuve

Friday 13/03/09 (11:00) [UCL, room FYMA - CYCL b328]

**Yukitaka Ishimoto** (Okayama Institute for Quantum Physics)

*"Solving infinite Kolam in knot theory"*

**Abstract:** In south India, many ladies draw a certain type of complicated line patterns every morning in front of their houses, with white rice powder on the ground. The class of pattern is drawn around a grid pattern of dots so that the lines minimally encircle each dot, which is called line "Kolam" pattern in Tamil.

Among them, one-line drawings or the "infinite Kolam" have some special meaning not only ethnologically but also mathematically. For example, we can address the following simple question: How many can we draw such patterns/diagrams for a given grid pattern of dots?

One may think that the problem is NP-hard alike: almost all cases should be examined for the solution. However, it is certainly not. In this talk, we focus on the knot-theoretical description of the infinite Kolam and show how to find the solution, which inevitably gives a sketch of the proof for the statement "infinite Kolam is not NP hard." Further discussions on Kolam may also be given.

Thursday 12/03/09 (16:30) [UCL, room FYMA - CYCL b328]

**Eric NORDENSTAM** (KTH - Stockholm)

*"The shuffling algorithm and the Aztec diamond"*

**Abstract:** The shuffling algorithm, introduced by Elkies et al., for sampling tilings of the Aztec diamond uniformly at random can be seen as a certain dynamics on a set of interacting particles. This is in some sense a discretization of a model of interlacing Brownian motions studied by Warren. As an application of these results, I will sketch a new proof of the fact that, in suitable scaling limit of large Aztec diamonds, one can recover the distribution of the eigenvalues of a GUE matrix and its principal minors.

Thursday 26/02/09 (10:45) [UCL, FYMA - CYCL b328]

**Carlangelo LIVERANI** (Université de Rome Tor Vergata)

*"Toward the derivation of the Fourier law from an Hamiltonian model"*

**Abstract**: I will discuss a program to obtain the Fourier law (or rather the heat equation), starting from an anharmonic lattice with some amount of noise. The noise is of a fixed level and it is not directly responsible for the energy exchanges. This is a joint work with Stefano Olla.

Thursday 12/02/09 (14:00) [UCL, FYMA - CYCL b328]

**Christian HAGENDORF** (Ecole Normale Supérieure, Paris)

*"Loop erased random walks and SLE"*

# Talks 2008

Tuesday 02/09/2008 (15:00) [KULeuven, room 02.18 Celestijnenlaan 200B]

**Pablo ROMAN** (Cordoba, Argentina)

*"Matrix valued spherical functions associated to the complex hyperbolic plane and the hypergeometric operator "*

Thursday 22/05/2008 (16:30) [UCL, Louvain-la-Neuve, Chemin du Cyclotron 2, CYCL 10]

**Patrik FERRARI** (Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin)

*"Anisotropic growth of random surfaces in 2+1 dimensions"*

Friday 25/04/2008 (16:00) [KULeuven, Celestijnenlaan 200B, Leuven (Heverlee), Room B.02.18]

**Alexander Its** (IUPUI, Indianapolis, USA)

*"Global asymptotic analysis of the Painlevé transcendents"*

Wednesday 23/04/2008 (10:30) [UCL, Louvain-la-Neuve, Chemin du Cyclotron 2, CYCL 10]

**Kurt Johansson** (KTH-Stockholm)

*"Universality results in random matrices"*

Friday 11th April at 14:00 - [Location : CYCL08 (Louvain La Neuve)]

**Micha Pevzner** (Universite de Champagne-Ardennes REIMS)

*"Crochets de Rankin-Cohen et Quantification des espaces symétriques"*

Thursday 3rd April at 16:00 - [Location : room CYCL08 at Louvain-la-Neuve, chemin du cyclotron]

**Steven Delvaux** (KULeuven)

*"A phase transition for non-intersecting Brownian motions, and the Painlevé II equation"*

# Talks 2007

Thursday 20/12/07 (16h15) [Heverlee, KULeuven, Celestijnenlaan 200B, salle 200B 01.18]

**Mark Adler** (Brandeis University)

*The Pearcey Process Revisited - An Attempt to Break Symmetry*

Thursday 06/12/2007 (16h15) [UCL, Louvain-la-Neuve, Chemin du Cyclotron 2, salle CYCL b328]

**Milton Jara** (Phys. UCL)

*An introduction to the tagged particle problem*

Thursday 22/11/07 (16h 15) [UCL, Louvain-la-Neuve, Chemin du Cyclotron 2, salle CYCL b328]

**Maurice Duits** [KULeuven-Math]

*Asymptotics of certain biorthogonal polynomials associated to a two-matrix model*

Thursday 08/11/2007 (16h15) [Heverlee, KULeuven, Celestijnenlaan 200S, salle 200S 00.04]

**Nicolas Orantin** (Math. UCL)

*Symplectic invariants and loop equations*

Thursday 25/10/2007 (16h15) [Lieu: Heverlee, KULeuven, Celestijnenlaan 200C, salle 200C 01.02]

**Nicolas Orantin** (Math. UCL)

*Formal matrix models and combinatorics of maps*

Thursday 11/10/2007 (16h15) [LLN - CYCL 10]

**Ivar Lyberg** (Phys. UCL)

*Form factor expansion of correlation functions of the two dimensional Ising lattice*

Friday 28/09/2007 (16h15) [ LLN - CYCL 02]

**Arno Kuijlaars** (Math. KULeuven)

*Non-intersecting squared Bessel processes*

Wednesday 19/09/2007 (16h30) [Heverlee, KULeuven, Celestijnenlaan 200 B, room 200B.02.18]

**Tom Claeys** (Math. KULeuven)

*The "birth of a cut"-transition in random matrix ensembles*

Friday 22/06/2007 (10h45) [LLN - CYCL04]

**Didier Vanderstichelen** (Math UCL)

*Modèles combinatoires et intégrales sur le groupe unitaire*

Friday 23/03/2007 (10h45) [LLN - CYCL04]

**Jean Bricmont ** (Physique UCL)

*Une introduction "pédestre" au groupe de renormalisation (2ème partie)*

Friday 09/03/2007 (10h 45) [LLN - CYCL04]

**Jean Bricmont ** (Physique UCL)

*Une introduction "pédestre" au groupe de renormalisation (1ère partie)*

Friday 23/02/2007 (10h 45) [LLN - CYCL04]

**Yuri Berest** (Cornell and IHES)

*Ideals of rings of differential operators on smooth curves*

# Talks 2006

Friday 08/12/2006 (10h45) [LLN - CYCL07]

**Augustin de Maere** (Phys. UCL)

*Transitions de phase dans les réseaux d'applications couplées II (preuve de l'existence)*

Friday 01/12/2006 (10h 45) [LLN - CYCL07]

**Augustin de Maere** (Phys. UCL)

*Transitions de phase dans les réseaux d'applications couplées I (exposé introductif)*

Friday 17/11/2006 (10h45) [LLN - CYCL07]

**Jonathan Delepine** (Math. UCL)

*Equations aux dérivées partielles pour la distribution du spectre de modèles matriciels aléatoires*

Friday 10/11/2006 (10h45) [LLN - CYCL07]

**Ariane Leblanc** (Math. UCL et Poitiers)

*Structures de (quasi)-Poisson quadratiques et construction d'un système intégrable sur un espace de modules*

Friday 03/11/2006 (10h45) [LLN - CYCL 07]

**François Huveneers** (Phys. UCL)

*Renormalisation et problèmes de petits diviseurs*

Friday 27/10/2006 (10h45) [LLN - CYCL07]

**Luc Haine** (Math. UCL)

*Collisions dans le système de Calogero-Moser*