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Affiche-annonce CHARLIER Christophe

IntégréTéléchargement
SECTEUR DES SCIENCES ET TECHNOLOGIES
INSTITUT DE RECHERCHE EN MATHEMATIQUE ET PHYSIQUE
Invitation à la soutenance publique de thèse
Pour l’obtention du grade de Docteur en Sciences
Monsieur Christophe CHARLIER
Master ingénieur civil en mathématiques appliquées à finalité spécialisée
Toeplitz and Hankel determinants in Random Matrix Theory
Toeplitz and Hankel determinants arise in many different areas of
mathematics, such as statistical mechanics and random matrix theory.
In Chapter 1, we review some historical aspects and known results on
these topics.
In Chapter 2 and Chapter 3, we study the determinants of Toeplitz
matrices as the size of the matrices tends to infinity, in the particular case
where the symbol is supported on the whole unit circle, has two jump
discontinuities and tends to zero at a certain rate on an arc of the unit
circle.
In Chapter 2, we find asymptotics for such Toeplitz matrices if the rate is
sufficiently fast. This generalizes a result proved by Widom, which was
known only for symbols supported on an arc of the unit circle.
In Chapter 3, we find asymptotics in the slower regime, which interpolates
between a symbol supported on an arc and a symbol with Fisher-Hartwig
singularities. This allows us to compute various probabilistic quantities in
the thinned and conditional Circular Unitary Ensemble. We study gap
probabilities for the thinned eigenvalues, and we study the statistics of the
eigenvalues of random unitary matrices which are conditioned such that
there are no thinned eigenvalues on a given arc of the unit circle.
In Chapter 4, we study the distribution of the ratio probability between the
smallest and second smallest eigenvalue in the Laguerre Unitary
Ensemble. We express this distribution as an integral of a Hankel
determinant. The limiting distribution as the size of the matrices tends to
infinity is found in terms of a function related to special solutions of a
system of ODEs which can be expressed in terms of a Riemann-Hilbert
problem.
Membres du jury :
Prof. Tom Claeys (UCL), promoteur
Prof. Luc Haine (UCL), promoteur
Prof. Michel Willem (UCL), président
Prof. Christian Hagendorf (UCL), secrétaire
Prof. Arno Kuijlaars (KULeuven)
Prof. Philippe Ruelle (UCL)
Prof. Gregory Schehr (Université Paris-Sud, France)
Lundi 4 juillet 2016 à 15h00
Auditoire CYCL01
Chemin du Cyclotron, 2
1348 Louvain-la-Neuve
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