Research interests
I am currently assistant professor (maître de conférences) at the Department of Mathematics of the University Paris-Sud (Orsay, France), and I am working in the team Probability and Statistics.
I am mainly interested in the two following parts of probability theory:
- convergence of random variables. With my collaborators (V. Féray, A. Nikeghbali, R. Chhaibi, … the list is growing!), we developed the theory of mod-ϕ convergence, which refines the central limit theorems and leads to large deviation principles, Berry-Esseen estimates, local limit theorems, etc. This theory allowed us to extend the classical probabilistic results on sums of i.i.d. to numerous other sequences of random variables, including: sums of random variables with a sparse dependency graph, characteristic polynomials of random matrices, functionals of a Markov chain, magnetisation of the Ising model, statistics of random graphs or permutations, arithmetic functions of a random integer.
- random objects on groups and symmetric spaces. I study random processes, random graphs and random combinatorial structures that are drawn on groups (finite, discrete or Lie) and their quotients, by using their (asymptotic) representation theory. These works are also connected to algebraic combinatorics, random matrix theory, free probability, and interacting systems of particles.
Recent works
Slides of a talk on fluctuations and concentration inequalities for central measures on integer partitions.
Slides of a recent talk on the spectrum of random geometric graphs on symmetric spaces.
Slides of another recent talk on mod-Gaussian convergence for graphon models.
Preliminary version of the memoir of my soon-to-be defended habilitation à diriger les recherches.
I usually put my papers on arXiv, but you can also look hereafter for my most recent pre-publications:
- Asymptotic representation theory and the spectrum of a random geometric graph on a compact Lie group, preliminary version which is almost complete. The first article of a long standing project that I started more than 3 years ago, and which consists in the study of the spectrum of a random geometric graphs on a symmetric space. This first paper treats the case of Lie groups, and it gives results for the Gaussian and the Poissonian regime, as well as conjectures regarding the limiting spectral measure in the Poissonian regime.
- Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces, with Valentin Féray and Ashkan Nikeghbali (preliminary version).
- Local limit theorems and mod-ϕ convergence, with Martina dal Borgo and Ashkan Nikeghbali. We prove local limit theorems for mod-ϕ convergent sequences of random variables, ϕ being a stable distribution. We give two new proofs of the local limit theorem in this framework: one proof based on the notion of zone of control, and one proof based on the notion of mod-φ convergence in L1(iR). These new approaches allow us to identify the infinitesimal scales at which the stable approximation is valid. We complete our analysis with a large variety of examples to which our results apply, and which stem from random matrix theory, number theory, combinatorics or statistical mechanics.
- Mod-ϕ convergence, III: Multi-dimensional mod-Gaussian convergence and related estimates of probabilities, with Valentin Féray and Ashkan Nikeghbali. We extend the results of the two previous papers to the setting of multi-dimensional mod-Gaussian convergence. This extension relies on some intricate geometrical arguments, which are required in order to control the convergence in law to the Gaussian distribution, and the precise moderate deviations of the multi-dimensional models.
Publications
My list of publications so far (click on a title to get the pdf):
- Mod-ϕ convergence: Approximation of discrete measures and harmonic analysis on the torus, with Reda Chhaibi, Freddy Delbaen and Ashkan Nikeghbali. Submitted.
- Mod-ϕ convergence, II: Estimates on the speed of convergence, with Valentin Féray and Ashkan Nikeghbali. To appear in Séminaire de Probabilités.
- Representation Theory of Symmetric Groups. Discrete Mathematics and Applications, 666+xvi p., CRC Press, 2017.
- Mod-ϕ convergence. Normality Zones and Precise Deviations, with Valentin Féray and Ashkan Nikeghbali. Springer Briefs in Probability and Mathematical Statistics, 152+xii p., Springer-Verlag, 2016.
- Mod-Gaussian convergence and its applications for models of statistical mechanics, with Ashkan Nikeghbali. In Memoriam Marc Yor - Séminaire de Probabilités XLVII, 369-425, LNM 2137, Springer-Verlag, 2015.
- Partial isomorphisms over finite fields, Journal of Algebraic Combinatorics, 40(1):83-136, 2014.
- The cut-off phenomenon for Brownian motions on compact symmetric spaces, Potential Analysis, 40(4):427-509, 2014.
- Fluctuations of central measures on partitions, Proceedings of the 24th International Conference on Formal Power Series and Algebraic Combinatorics (Nagoya, Japan), p. 387-398, 2012.
- Kerov’s central limit theorem for Schur-Weyl and Gelfand measures, Proceedings of the 23rd International Conference on Formal Power Series and Algebraic Combinatorics (Reykjavík, Iceland), p. 669-680, 2011.
- Products of Geck-Rouquier conjugacy classes and the algebra of composed permutations, Proceedings of the 22nd International Conference on Formal Power Series and Algebraic Combinatorics (San Francisco, USA), p. 789-800, 2010.
- Asymptotics of q-Plancherel measures, with Valentin Féray, Probability Theory and Related Fields, 152(3-4):589-624, 2012.
Academic works
I did my Ph. D. in 2007-2010 under the supervision of Philippe Biane, at the University Paris-Est Marne-la-Vallée (France); here is the dissertation: