Personal Webpage of Quentin FRANCOIS

I am a PhD student in Probability theory under the supervision of Djalil Chafaï and Pierre Tarrago. I am mostly interested in high dimensional phenomenas arising from random matrix theory and exactly solvable models.

Research Interests: Random Matrices, Free Probability, Integrable Probability and Algebraic Combinatorics.

Contact: firstname.lastname@dauphine.psl.eu

Office(s):

B223, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris
V3, École Normale Supérieure, 45 rue d’Ulm, 75005 Paris

Papers:

Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble.

Q. François, and D. Garcìa-Zelada

2023

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Abstract: We consider the complex Elliptic Ginibre Ensemble, a family of random matrix models introduced by Girko that interpolates between the Ginibre Ensemble and the Gaussian Unitary Ensemble and such that its empirical spectral measure converges to the uniform measure on an ellipse. We show the convergence in law of its normalised characteristic polynomial outside of this ellipse. Our proof contains two main steps. We first show the tightness of the normalised characteristic polynomial using the link between the Elliptic Ginibre Ensemble and Hermite polynomials. This part relies on the uniform control of the Hermite kernel which is derived from the recent work of Akemann, Duits and Molag. In the second step, we identify the limiting object as the exponential of a Gaussian analytic function. The limit expression is derived from the convergence of traces of random matrices, based on an adaptation of techniques that were used to study fluctuations of Wigner and deterministic matrices by Male, Mingo, Péché and Speicher. This work answers the interpolation problem raised in the work of Bordenave, Chafaï and the second author of this paper for the integrable case of the Elliptic Ginibre Ensemble and is therefore a fist step towards the conjectured universality of this result.
Positive formula for the product of conjugacy classes on the unitary group.

Q. François, and P. Tarrago

2024

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Abstract: The convolution product of two conjugacy classes of the unitary group U_n is described by a probability distribution on the space of central measures. Relating this convolution to the quantum cohomology of Grassmannians and using recent results describing the structure constants of the latter, we give a manifestly positive formula for the density of the probability distribution for the product of generic conjugacy classes. In the same flavor as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. As a consequence, this expression also provides a positive and explicit formula for the volume of SU_n-valued flat connections on the three-holed two dimensional sphere, which was first given by Witten in terms of an infinite sum of characters.
Enumeration of crossings in two-step puzzles.

Q. François

2024

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Abstract: We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.
Characteristic polynomial of generalized Ewens random permutations.

Q. François

2025

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Abstract: We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure, according to certain weights assigned to each cycle length. The proof is based on uniform control of the characteristic polynomial using results from the singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which has already appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian Holomorphic Chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.

Research Talks:

Working Group: Point Processes and Applications, Febuary 7, 2025, Laboratoire Paul-Painlevé, Lille, France.
Young Researcher Seminar, December 19, 2024, Université Paris-Dauphine, Paris, France.
Probability and Statistics seminar, November 22, 2024, Laboratoire de Mathématiques de Versailles, Versailles, France.
Les Probas du Vendredi, October 4, 2024, Sorbonne Université, Paris, France.
Fifth meeting of ANR CORTIPOM, October 1, 2024, Institut Denis Poisson, Tours, France.
Young Researcher Days, June 12, 2024, Calvados, France.
Seminar Series of the Long Program: Geometry, Statistical Mechanics and Integrability, May 16, 2024, Institute for Pure and Applied Mathematics UCLA, Los Angeles, USA.
MEGA Seminar, January 12, 2024, IHP Paris, France.
Young Researcher Seminar,
December 14, 2023, Université Paris-Dauphine, Paris, France.
Combinatoire Elliptique et au delà, November 16, 2023, Institut d’Études Scientifiques de Cargèse, Corse, France.
Probability and Statistics Seminar, November 7, 2023, Institut de Mathématiques de Marseille, Marseille, France.
Colloquium for PhD students, October 25, 2023, École Normale Supérieure, Paris, France.
Image, Optimization and Probability Seminar, September 21, 2023, Institut Mathématique de Bordeaux, Bordeaux, France.
Young Researcher Days, June 1, 2023, Université Paris-Dauphine, Paris, France.
Young Researcher Seminar, March 3, 2023, Université Paris-Dauphine, Paris, France.

Conference Attendance:

2025

Géométries aléatoires et applications, January 28-29, Collège de France, Paris, France.

2024

Algebraic Aspects of Random Matricesi, October 7-11, CIRM, Luminy, France.
Fifth meeting of ANR CORTIPOM, September 30 - October 2, Institut Denis Poisson, Tours, France.
Young Researcher Days, June 10-12, Calvados, France.
Long Program: Geometry, Statistical Mechanics and Integrability, March 11 - May 30, Institute for Pure and Applied Mathematics UCLA, Los Angeles, USA.
Fourth meeting of ANR CORTIPOM, January 18-19, Sorbonne Université, Paris, France.

2023

Combinatoire elliptique et au delà !, November 13-17, Institut d’Études Scientifiques de Cargèse, Corse, France.
CORTIPOM project workshop, July 10-14, CIRM, Luminy, France.
Third PIICQ Workshop, May 22-26, Angers, France.

Teaching and Supervision:

2024-2025

Master Thesis supervision: Limit Theorems for large random matrices, Université Paris-Dauphine.
Introduction aux Probabilités (Bachelor course, Université Paris-Dauphine). Teaching Assistant.

2023-2024

Probabilités I (Bachelor course, Université Paris-Dauphine). Teaching Assistant.
Processus Discrets (Master course, Université Paris-Dauphine) Teaching Assistant.

2022-2023

Probabilités I (Bachelor course, Université Paris-Dauphine). Teaching Assistant.
Processus Discrets (Master course, Université Paris-Dauphine). Teaching Assistant.