(université de
Genève)
The seminar usually takes place on mondays at 16:15 in room 17
Lundi 04 mars 2013: pas de séminaire.
Lundi 11 mars 2013: Gaetan Borot
(Université de Genève) All order asymptotics of beta
ensembles in the multi-cut regime
The beta ensemble is a statistical-mechanical model of N particles
x_i on the real line, subjected to pairwise Coulomb repulsion, and
trapped in an external potential. In particular cases, this model
describes the distribution of eigenvalues of random matrices. Old
results from potential theory implies under weak assumptions that
the random distribution of x_i converges almost surely and in
expectation towards a deterministic distribution, supported on a
collection of (g + 1)-segments. The fluctuations around this
distribution depend much on the topology of the support, i.e. the
number g. When g = 1 (the one-cut regime), Johansson established in
1998 that the linear statistics h(x_1) + ... + h(x_N) satisfy a
central limit theorem for smooth test functions h, and the partition
function and expectation values of multilinear statistics are
expected to have a 1/N expansion. When g > 1, this is not anymore
true in general, the partition function and multilinear statistics
should feature a pseudo-periodic behavior in N at all orders in a
1/N expansion. This was predicted by Bonnet, David, Eynard in 2000
at leading order, and refined by Eynard in 2007.
I will present a joint work with Alice Guionnet, where we justify
rigorously those heuristics, and establish all-order expansions in
beta ensembles for real-analytic potentials are away from critical
points. Since random matrices are related to integrable systems and
orthogonal polynomials, this allows us to establish all-order
asymptotic expansion of certain solutions of the Toda chain in the
continuum limit and multi-cut regime, and orthogonal polynomials and
skew-orthogonal polynomials away from the bulk. The methods are
purely probabilistic and provide an alternative to Riemann-Hilbert
techniques, with some advantages and inconvenients I will try to
explain.
Lundi 18 mars 2013 Sacha Glazman (Université de Genève) TBA
Lundi 25 mars 2013: Juan Rivera-Letelier (Pontificia Universidad Catolica de Chile) Low-temperature phase transitions in the quadratic family
We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense. This is a joint work with Daniel Coronel.
Lundi 1er avril 2013 vacances de Paques.
Lundi 8 avril 2013 TBA.
Lundi 15 avril 2013 TBA.
Lundi 22 avril 2013 TBA.
Lundi 29 avril 2013 TBA.
Lundi 6 mai 2013: Omer Angel (UBC)
Lundi 13 mai 2013: Laure Dumas (ENS ulm)
If you have any question, you can reach me at 0041223791169
