PAPERS
Powers of Ginibre Eigenvalues
(Prepublished on November 8th, 2017.)
The distribution of overlaps between eigenvectors of Ginibre matrices
(Joint work with P. Bourgade. Prepublished on January 3rd, 2018.)
This video illustrates one aspect of our results.
TALKS
I gave a Junior Participant talk at the 2017 NorthEast Probability Seminar.
(November 16th).
Title: The overlaps between Ginibre eigenvectors
Abstract : The Complex Ginibre Ensemble consists of matrices whose coefficients are i.i.d. complex Gaussian. The eigenvectors are nonorthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dysontype dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of the overlaps and their correlations. (Joint work with P. Bourgade)
STUDENT PROBABILITY SEMINAR
FALL 2017
The Student
Probability Seminar happens on wednesdays, 10am to 11am, usually in room 1314.
Student Probability Seminar,
october 18th.
At 10am in room WWH 1314.
Title : SelfAvoiding Walks on the Honeycomb Lattice.
Abstract : While in general there is no exact formula that counts
selfavoiding walks on a given infinite graph, the number of selfavoiding
walks of length k on lattices is known to be logarithmically
equivalent to c^{k} , where c is called the connective
constant of the lattice, and usually can only be approximated. In the
specific case of the honeycomb lattice (i.e. the hexagonal structure one
finds in beehives), physical heuristics led to the conjecture that the
connective constant is
DuminilCopin and
Smirnov proved this conjecture a few years ago in a sensational
thirteenpagelong paper. We will sketch their proof, discuss why it cannot
be easily extended to other settings, and what else can be asked or
expected from a probabilistic point of view.
SPRING 2017
Student Probability Seminar,
april 3rd.
At 11am in room WWH 905.
Title : Ginibre Powers  the Laundry Machine Effect.
Abstract : The name of Jean Ginibre was given to a very natural ensemble of random matrices, those with iid complex gaussian coefficents. The eigenvalues of such a matrix form a determinantal point process, and thus exhibit a wellstudied repulsion. But this highly correlated system of points happens to decorrelate completely (I say : completely  not asymptotically) when put to a high enough power  that is, having them spin around the origin. This stunning property is known to hold for a wider class of processes; but the proof is particularly straightforward in the Ginibre case, as we shall see.
Student Probability Seminar,
march 6th.
At 11am in room WWH 102.
Title : Some problems of random cycles  and how to fix them.
Abstract : We shall consider the uniform distribution over the
permutation groups and try to answer simple questions such as : how likely
is it for two elements to belong to the same cycle ? How are fixed points
distributed ? What is the typical size of a cycle ?... Some of our answers
will be just as simple as the questions; whereas others will require to
introduce usual tools and methods of random permutation theory, such as the
celebrated Feller coupling, thanks to which we might even end up answering
questions we didn't ask.
FALL 2016
Student Probability Seminar,
october 25th.
At 2pm in room WWH 1314.
Title : The Characteristic Polynomial of a Random Unitary Matrix.
Abstract : We will introduce the Circular Unitary Ensemble (CUE) of random unitary
matrices distributed according to the Haar measure on U_{n}(C), and compute the
distribution of its characteristic polynomial thanks to an explicit decomposition of
the law. In particular, the moments of this characteristic polynomial can be
computed ; they are strongly believed to have a link with the moments of the zeta
function along its critical line, and we will give some reasons why this might be
true.
SPRING 2016
Student Probability Seminar,
february 25th.
At 4pm in room WWH 1314.
Title : Mere coincidences,
or perhaps more.
Abstract : The story of the meeting at tea
time between Freeman Dyson and Hugh
Montgomery, and what happened that the
latter would later recall as 'real
serendipity', is now famous; but a good
story never bores. We shall tell it again,
sketch its mathematical background, and
present a few other numbertheoretic objects
that exhibit amazing similarities with
wellknown results in Random Matrix theory :
not only the zeros of the Zeta function, but
also its moments on the critical line, andthe
number of points of some elliptic curves over
finite fields.
FALL 2015
Student Probability Seminar,
october 29th, 2015.
Title : Around Ulam's
problem.
Abstract : This talk will recall and sketch
famous results about the longest increasing
subsequence of a random (uniformly chosen)
permutation. For this purpose,we
shall learn and play a solitaire game, give
a probabilistic proof of two elegant
hookformulas, and explain the
RobinsonSchensted algorithm, among other
things. These tools will give us two
different approaches of the same problem,
and even allow us to draw some conclusions.
I took and passed my Oral
examination on october 13th.
Here is
my outline.
BEFORE 2015
My Master's Thesis "Approches
probabilistes de la fonction Zêta" will be
displayed online one day.
My Mémoire de Maîtrise, "Corps
de uinvariant pair" was cowritten
with Margaret Bilu. The uinvariant of a field k
is the maximal dimension of an anisotropic
quadratic form on k. It was conjectured at
some point that it should be a power of 2 for
every field; which is not the case. In this paper,
following an argument given by A. Merkurjev, we
build a field of uinvariant 2n for any
positive integer n.


