## PhD thesis

Processes on the unitary group and on the linear group and free probability.

## HDR thesis

Probabilités non-commutatives et matrices aléatoires.

**Combinatorics of cyclic-conditional freeness** with Octavio Arizmendi and Nicolas Gilliers

*arXiv:2311.13178, 2023.* [Abstract] [pdf]
We develop the relevant combinatorics pertaining to cyclic-conditional freeness in order to introduce the adequate sets of cumulants linearizing the cyclic-conditional additive convolution. On our way, we introduce a new non-commutative independence, cyclic freeness. We explain how cyclic-conditional freeness is "reduced" to cyclic freeness by utilizing a multivariate extension of the inverse Markov-Krein transform. We also consider cyclic-conditional multiplicative convolution and prove limit theorems.

**Freeness of type B and conditional freeness for random matrices** with Antoine Dahlqvist and Franck Gabriel

*Indiana University Mathematics Journal, Indiana Univ. Math. J., Vol. 73 (No. 03), 2024.* [Abstract] [pdf]
The asymptotic freeness of independent unitarily invariant NxN random matrices holds in expectation up to O(1/N^2). An already known consequence is the infinitesimal freeness in expectation. We put in evidence another consequence for unitarily invariant random matrices: the almost sure asymptotic freeness of type B. As byproducts, we recover the asymptotic cyclic monotonicity, and we get the asymptotic conditional freeness. In particular, the eigenvector empirical spectral distribution of the sum of two randomly rotated random matrices converges towards the conditionally free convolution. We also show new connections between infinitesimal freeness, freeness of type , conditional freeness, cyclic monotonicity and monotone independence. Finally, we show rigorously that the BBP phase transition for an additive rank-one perturbation of a GUE matrix is a consequence of the asymptotic conditional freeness, and the arguments extend to the study of the outlier eigenvalues of other unitarily invariant ensembles.

**Universality of free random variables: atoms for non-commutative rational functions** with Octavio Arizmendi, Roland Speicher and Sheng Yin

*Advances in Mathematics, Volume 443, 2024.* [Abstract] [pdf]
We address the following question: what can one say, for a tuple (Y1, . . . , Yd)
of normal operators in a tracial operator algebra setting with prescribed sizes of the
eigenspaces for each Yi, about the sizes of the eigenspaces for any non-commutative
polynomial P (Y1, . . . , Yd) in those operators? We show that for each polynomial P there
are unavoidable eigenspaces, which occur in P (Y1, . . . , Yd) for any (Y1, . . . , Yd) with the
prescribed eigenspaces for the marginals. We will describe this minimal situation both
in algebraic terms - where it is given by realizations via matrices over the free skew
field and via rank calculations - and in analytic terms - where it is given by freely
independent random variables with prescribed atoms in their distributions. The fact
that the latter situation corresponds to this minimal situation allows to draw many
new conclusions about atoms in polynomials of free variables. In particular, we give
a complete description of atoms in the free commutator and the free anti-commutator.
Furthermore, our results do not only apply to polynomials, but much more general also
to non-commutative rational functions.

**Asymptotic cyclic-conditional freeness of random matrices** with Nicolas Gilliers

*Random Matrices: Theory and Applications, Vol. 13 (No. 01), 2024.* [Abstract] [pdf]
Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on N×N random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix U_N. In this article we elaborate on the previous point by proposing a random matrix model, which we name the Vortex model, where UN has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector v_N. In the limit N→+∞, we show that N×N matrices randomly rotated by the matrix U_N are asymptotically conditionally free with respect to the normalized trace and the state vector v_N. To describe second order asymptotics, we define cyclic-conditional freeness, a new notion of independence unifying infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. The infinitesimal distribution in the Vortex model can be computed thanks to this new independence. Finally, we elaborate on the Vortex model in order to build random matrix models for ordered freeness and for indented independence.

** Universal constructions for spaces of traffics ** with Antoine Dahlqvist and Camille Male

*Documenta Mathematica, Volume 29 (No. 1), 2024.* [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.

** Quantum groups based on spatial partitions ** with Moritz Weber

*Annales de la Faculté des Sciences de Toulouse, Vol.32 (No.4), 727-768, 2023.* [Abstract] [pdf]
We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions -- which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang's free orthogonal quantum group O+n which do not contain the symmetric group Sn; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of certain categories of partitions which do neither contain the pair partition nor the identity partition.

**Operator-Valued Matrices with Free or Exchangeable Entries ** with Marwa Banna

*Annales de l'IHP Probabilités et statistiques, Vol.59 (No.1), 503–537, 2023.* [Abstract] [pdf]
We study matrices whose entries are free or exchangeable noncommutative elements in some tracial W∗-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and prove quantitative convergence to operator-valued semicircular elements over some subalgebra in terms of Cauchy transforms. As direct applications, we obtain explicit rates of convergence for a large class of random block matrices with independent or correlated blocks. Our approach relies on a noncommutative extension of the Lindeberg method and operator-valued Gaussian interpolation techniques.

**De Finetti Theorems for the unitary dual group** with Isabelle Baraquin, Uwe Franz, Laura Maaßen and Moritz Weber

*SIGMA, Vol.18 (No.67), 2022.* [Abstract] [pdf]
We prove several de Finetti Theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti Theorem characterizing R-diagonal elements with identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti Theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in W*-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group. Thirdly, the above de Finetti Theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti Theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in W*-probability spaces. On the other hand, if we drop the assumption of faithful states in W*-probability spaces, we obtain a non-trivial half a de Finetti Theorem similar to the case of the dual group action.

** Fluctuations of Brownian Motions on GL_N ** with Todd Kemp

*Ann. Inst. Henri Poincaré Probab. Stat. 58 (1), 524-547, 2022.* [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.

**Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices ** with Serban Belinschi, Charles Bordenave and Mireille Capitaine

*Electronic Journal of Probability 2021, Vol. 26, paper no. 100, 1-37.* [Abstract] [pdf]
We consider a square random matrix of size N of the form P(Y,A) where P is a noncommutative polynomial, A is a tuple of deterministic matrices converging in ∗-distribution, when N goes to infinity, towards a tuple a in some C∗-probability space and Y is a tuple of independent matrices with i.i.d. centered entries with variance 1/N. We investigate the eigenvalues of P(Y,A) outside the spectrum of P(c,a) where c is a circular system which is free from a. We provide a sufficient condition to guarantee that these eigenvalues coincide asymptotically with those of P(0,A).

**Large permutation invariant random matrices are asymptotically free over the diagonal ** with Benson Au, Antoine Dahlqvist, Franck Gabriel and Camille Male

*Annals of Probability, Vol. 49, No. 1, 157-179, 2021.* [Abstract] [pdf]
We prove that independent families of permutation invariant random matrices are
asymptotically free over the diagonal, both in probability and in expectation, under a uniform
boundedness assumption on the operator norm. We can relax the operator norm
assumption to an estimate on sums associated to graphs of matrices, further extending the
range of applications (for example, to Wigner matrices with exploding moments and so the
sparse regime of the Erd\H {o} sR\'{e} nyi model). The result still holds even if the matrices
are multiplied entrywise by bounded random variables (for example, as in the case of
matrices with a variance profile and percolation models).

** A quantitative fourth moment theorem in free probability theory**

*Advances in Mathematics, Volume 380, 2021, 107579.* [Abstract] [pdf]
A quantitative" fourth moment theorem" is provided for any self-adjoint element in a
homogeneous Wigner chaos: the Wasserstein distance is controlled by the distance from the
fourth moment to two. The proof uses the free counterpart of the Stein discrepancy. On the
way, the free analogue of the WSH inequality is established.

**A note on existence of free Stein kernels ** with Max Fathi and Tobias Mai

*Proceedings of the AMS 148: 1583-1594, 2020.* [Abstract] [pdf]
Stein kernels are a way of comparing probability distributions, defined via integration by parts formulas. We provide two constructions of Stein kernels in free probability. One is given by an explicit formula, and the other via free Poincaré inequalities. In particular, we show that unlike in the classical setting, free Stein kernels always exist. As corollaries, we derive new bounds on the rate of convergence in the free CLT, and a strengthening of a characterization of the semicircular law due to Biane.

** Fluctuation of matrix entries and application to outliers of elliptic matrices ** with Florent Benaych-Georges and Jean Rochet

*Canad. J. Math. 70, 3-25, 2018.* [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.

**Segal-Bargmann transform: the q-deformation** with Ching-Wei Ho

*Lett Math Phys 108: 1677, 2018.* [Abstract] [pdf]
We give identifications of the q-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed q-Gaussian variables. We prove that, when defined on the random matrix model of \'Sniady for the q-Gaussian variable, the classical Segal-Bargmann transform converges to the q-deformed Segal-Bargmann transform in the large N limit. We also show that the q-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.

** The generalized master fields ** with Antoine Dahlqvist and Franck Gabriel

*Journal of Geometry and Physics, 119, pp. 34–53, 2017.* [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.

** Haar states and Lévy processes on the unitary dual group ** with Michael Ulrich

*Journal of Functional Analysis, 270 (7), pp. 2769-2811, 2016.* [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.

** Matricial model for the free multiplicative convolution **

*Annals of Probability, 44 (4), pp. 2427 - 2478, 2016.* [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.

** Free convolution operators and free Hall transform **

*Journal of Functional Analysis, 265 (11), pp. 2645-2708, 2013.* [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.