## ANR JCJC: Space of Traffics and Asymptotics of Random Spectra (STARS)

**Project members**

Mireille Capitaine, Guillaume Cébron, Francois Chapon, Reda Chhaibi, Antoine Dahlqvist, Franck Gabriel, Cécilia Lancien, Camille Male, Pascal Maillard and Ion Nechita.

**Scientific project**

Random Matrix Theory developed during the last three decades in numerous fields of mathematics and physics. Free Probability Theory (FPT) is adapted for their analysis in large dimension. Operator-valued free probability and Traffic Probability Theory (TPT) are upgrades of FPT appropriate to describe random matrices which are beyond the usual scope of FPT.

This project aims at fostering new directions of research in the fields of FPT and TPT, via the study of random matrices subject to new symmetries and via new applications. We will investigate questions such as the convergence of the spectrum distributions and the understanding of the outliers, in interaction with the invariances of the random matrix models. Fields of application we shall explore include Random Matrix Theory, Lie group theory, signal detection theory and artificial neural networks.

Our goals are divided into the four following axes: Stieltjes transform methods with amalgamation, Schur-Weyl duality and extension to other invariances, Signal-plus-noise matrix models, and Neural networks.

You can find some references at the end of this webpage.

**Post-doctoral position**

Applications are invited for a postdoctoral position at the Mathematics Institute of Toulouse, funded by the ANR project STARS. The position is for two years, with a starting date between 01/04/21 and 01/01/22. All candidates interested in working on one of the following topics are encouraged to apply: random matrix theory, traffic probability theory, free probability theory, and their interplay with signal detection and neural networks. Applications, including a CV, a list of publications and a research statement should be sent to Guillaume Cébron ( guillaume.cebron@math.univ-toulouse.fr ). Applicants should also arrange for one letter of recommendation to be sent to the same address. They will be evaluated first after February 28th, and then on a rolling basis until the position is filled. Requests for information about the position should also be sent to the same address.

**Master scholarships**

we offer Master grants in mathematics as part of its action program. We expect to supervise students from the Master Research and Innovation in Toulouse, but alos outside students interested in working on random matrices. Do not hesitate to contact one of the project member.

**Some references related to this project**

__Traffic probability theory:__

**Traffic distributions and independence I: permutation invariant random matrices and the three notions of independence**[pdf] by Male

*To appear in Memoirs of the AMS*

**Traffic distributions and independence II: Universal constructions for traffic spaces**[pdf] by Cébron, Dahlqvist and Male

**Large permutation invariant random matrices are asymptotically free over the diagonal**[pdf] by Au, Cebron, Dahlqvist, Gabriel and Male

*To appear in Annals of Probability*

**Rigid structures in the universal enveloping traffic space**[pdf] by Au and Male

__Free probability theory:__

**Deformed ensembles, polynomials in random matrices and free probability theory**[pdf] by Capitaine

*Habilitation à Diriger des Recherches*

**Free probability and Random matrices**[pdf] by Mingo and Speicher

*Fields Institute Monographs, Springer*

**Lectures on the combinatorics of free probability**[pdf] by Nica and Speicher

*London Mathematical Society Lecture Note Series, Vol. 335, Cambridge University Press*

**Blog on "Free Probability"**[link] by Speicher