# Séminaires GPP

Louvain-La-Neuve

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 Duschene (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.