Research interests: Random matrix theory and free probability

Curriculum vitae

Mini-workshop "Notions of freeness" April 11 - April 15, 2016

Articles (see also Google scholar profile):

2016

• Fluctuation of matrix entries and application to outliers of elliptic matrices ,
  with Florent Benaych-Georges and Jean Rochet
  ArXiv:1602.02929. [Abstract] [pdf]

  For any family of $N\times N$ random matrices $(\mathbf{A}_k)_{k\in K}$ which is invariant, in law, under unitary conjugation, we give general conditions for central limit theorems of random variables of the type $\operatorname{Tr}(\mathbf{A}_k \mathbf{M})$, where the Euclidean norm of $\mathbf{M}$ has order $\sqrt{N}$ (such random variables include for example the normalized matrix entries $\sqrt{N} \mathbf{A}_k(i,j)$). A consequence is the asymptotic independence of the projection of the matrices $\mathbf{A}_k$ onto the subspace of null trace matrices from their projections onto the orthogonal of this subspace. This result is used to study the asymptotic behaviour of the outliers of a spiked elliptic random matrix. More precisely, we show that their fluctuations around their limits can have various rates of convergence, depending on the Jordan Canonical Form of the additive perturbation. Also, some correlations can arise between outliers at a macroscopic distance from each other. These phenomena have already been observed by Benaych-Georges and Rochet with random matrices from the Single Ring Theorem.

• The generalized master fields with Antoine Dahlqvist and Franck Gabriel
  ArXiv:1601.00214. [Abstract] [pdf]

  The master field is the large $N$ limit of the Yang-Mills measure on the Euclidean plane. It can be viewed as a non-commutative process indexed by paths on the plane. We construct and study generalized master fields, called free planar Markovian holonomy fields, which are versions of the master field where the law of a simple loop can be as more general as it is possible. We prove that those free planar Markovian holonomy fields can be seen as well as the large $N$ limit of some Markovian holonomy fields on the plane with unitary structure group.

• Universal constructions for spaces of traffics with Antoine Dahlqvist and Camille Male
  ArXiv:1601.00168. [Abstract] [pdf]

  We investigate questions related to the notion of traffics introduced by the author C. Male as a noncommutative probability space with numerous additional operations and equipped with the notion of traffic independence. We prove that any sequence of unitarily invariant random matrices that converges in noncommutative distribution converges in distribution of traffics whenever it fulfills some factorization property. We provide an explicit description of the limit which allows to recover and extend some applications (on the freeness from the transposed ensembles by Mingo and Popa and the freeness of infinite transitive graphs by Accardi, Lenczewski and Salapata). We also improve the theory of traffic spaces by considering a positivity axiom related to the notion of state in noncommutative probability. We construct the free product of spaces of traffics and prove that it preserves the positivity condition. This analysis leads to our main result stating that every noncommutative probability space endowed with a tracial state can be enlarged and equipped with a structure of space of traffics.

2015

• Haar states and Lévy processes on the unitary dual group with Michael Ulrich
  Journal of Functional Analysis, 270 (7), pp. 2769-2811 [Abstract] [pdf]

  We study states on the universal noncommutative *-algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. Finally, we study a particular class of free L\'evy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity.

2014

• Fluctuations of Brownian Motions on GL_N with Todd Kemp
  To appear in Ann. Inst. Henri Poincaré Probab. Stat. [Abstract] [pdf]

  We consider a two parameter family of unitarily invariant diffusion processes on the general linear group $\mathbb{GL}_N$ of $N\times N$ invertible matrices, that includes the standard Brownian motion as well as the usual unitary Brownian motion as special cases. We prove that all such processes have Gaussian fluctuations in high dimension with error of order $O(1/N)$; this is in terms of the finite dimensional distributions of the process under a large class of test functions known as trace polynomials. We give an explicit characterization of the covariance of the Gaussian fluctuation field, which can be described in terms of a fixed functional of three freely independent free multiplicative Brownian motions. These results generalize earlier work of L\'evy and Ma\"ida, and Diaconis and Evans, on unitary groups. Our approach is geometric, rather than combinatorial.

• Matricial model for the free multiplicative convolution
  To appear in Annals of Probability. [Abstract] [pdf]

  This paper investigates homomorphisms \`a la Bercovici-Pata between additive and multiplicative convolutions. We also consider their matricial versions which are associated with measures on the space of Hermitian matrices and on the unitary group. The previous results combined with a matricial model of Benaych-Georges and Cabanal-Duvillard allows us to define and study the large N limit of a new matricial model on the unitary group for free multiplicative L\'evy processes.

2013

• Free convolution operators and free Hall transform
  Journal of Functional Analysis, 265 (11), pp. 2645-2708 [Abstract] [pdf]

  We define an extension of the polynomial calculus on a W*-probability space by introducing an abstract algebra which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal-Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N tends to infinity, of the *-distribution of the Brownian motion on the linear group GL_N(C) to the *-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N) to the free Hall transform.

Phd thesis: Processus sur le groupe unitaire et probabilités libres (2014).