Complex Ginibre matrices are random matrices with i.i.d. complex Gaussian entries. It is well known that their
eigenvalues exhibit quadratic repulsion, and that their empirical distribution converges to the circular law (uniform
distribution on the unit disk) when properly scaled. Although these eigenvalues are strongly correlated, their images
under power maps can be analyzed as the superposition of several independent blocks with explicit joint densities.
The proof technique that we will present extends to other distributions, and allows to recover Kostlan's theorem : the
radii of complex Ginibre eigenvalues are distributed like independent points
Eigenvectors of non-hermitian matrices are non-orthogonal, 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 Dyson-type dynamics. They first appeared in the physics literature; 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.
Abstract : The Complex Ginibre Ensemble consists of matrices whose coefficients are i.i.d. complex Gaussian. The eigenvectors are non-orthogonal, 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 Dyson-type 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)
I have been co-organizing the student probability seminar for three years and gave the following talks.
Several concepts and results of Random Matrix Theory also have a representation theoretic interpretation. The purpose of this talk is to show that this connection is not as esoteric as it sounds. We will introduce the basic concepts of representation theory from scratch, and see a few ways in which characters and representations of the unitary and symmetric groups appear in a random matrix context. In particular, our goal will be to understand an elegant proof of a Central Limit Theorem by Diaconis and Shahshahani that relies on such ideas.
Abstract : While in general there is no exact formula that counts
self-avoiding walks on a given infinite graph, the number of self-avoiding
walks of length k on lattices is known to be logarithmically
equivalent to ck , 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
Smirnov proved this conjecture a few years ago in a sensational
thirteen-page-long 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.
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 well-studied 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.
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.
Abstract : We will introduce the Circular Unitary Ensemble (CUE) of random unitary
matrices distributed according to the Haar measure on Un(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
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 'realserendipity', is now famous; but a good story never bores. We shall tell it again,sketch its mathematical background, and present a few other number-theoretic objects that exhibit amazing similarities with well-known results in Random Matrix theory :not only the zeros of the Zeta function, but also its moments on the critical line, and the number of points of some elliptic curves over finite fields.
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 hook-formulas, and explain the Robinson-Schensted algorithm, among other things. These tools will give us two different approaches of the same problem, and even allow us to draw some conclusions.
► My Mémoire de Maîtrise,"Corps
de u-invariant pair" was co-written
with Margaret Bilu. The u-invariant 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 u-invariant 2n for any
positive integer n.