Research

Research interests

I am interested in high dimensional phenomena occurring in probability, graph theory, convex geometry, and functional analysis. My research areas include: Random matrices, random graphs, concentration inequalities, Markov chains mixing time, Geometric functional analysis, convex geometry, functional inequalities, quantum information theory, mathematical foundation of machine learning.

Publications

• Graphon-Theoretic approach to central limit theorems for ϵ-independence, arXiv:2411.13062.
  Joint with Guillaume Cébron and Patrick Oliveira Santos.
  
• Finite-Sample Analysis of the Monte Carlo Exploring Starts Algorithm for Reinforcement Learning, arXiv:2410.02994.
  Joint with Suei-Wen Chen and Keith Ross.
  
• Central limit theorem for tensor products of free variables, arXiv:2404.19662. 
  Joint with Cécilia Lancien and Patrick Oliveira Santos.
  
• How bad is training on synthetic data? A statistical analysis of language model collapse, COLM 2024,  arXiv:2404.05090. 
  Joint with Mohamed Seddik, Suei-Wen Chen, Soufiane Hayou, Merouane Debbah. 
• On spectral outliers of inhomogeneous symmetric random matrices, arXiv:2401.07852.
  Joint with Dylan J.Altschuler, Patrick Oliveira Santos, and Konstantin Tikhomirov. 
• Limiting spectral distribution of random self-adjoint quantum channels, to appear in Mathematical Physics, Analysis, and Geometry (MPAG), arXiv:2311.12368.
  Joint with Cécilia Lancien and Patrick Oliveira Santos. 
• Flat bands of periodic graphs, J. Math. Phys. 64, 092101 (2023), arXiv:2304.06465.
  Joint with Mostafa Sabri.
• A note on Quantum Expanders, arXiv:2302.07772.
  Joint with Cécilia Lancien.
  
• Upgrading MLSI to LSI for reversible chains, J. Funct. Anal. 285 (9):110076,2023, arXiv:2212.06028. 
  Joint with Justin Salez and Konstantin Tikhomirov. 
• Monotonicity of the logarithmic energy for random matrices, to appear in Random Matrices: Theory and Applications, arXiv:2212.06090.
  Joint with Djalil Chafaï and Benjamin Dadoun. 
• Regularized modified log-Sobolev inequalities, and comparison of Markov chains, to appear in Annals of Probability, arXiv:2206.12477.
  Joint with Konstantin Tikhomirov.

• Maximal correlation and monotonicity of free entropy, Electron. Commun. Probab. 26: 1-10 (2021), arXiv:2011.03045.
  Joint with Benjamin Dadoun.

• Sharp Poincare and log-Sobolev inequalities for the switch chain on regular bipartite graphs, Probab. Theory Relat. Fields (2022),  arXiv:2007.02729.
  Joint with Konstantin Tikhomirov.
• Matrix Poincaré inequalities and concentration, Adv. Math., 371:107251, 2020, arXiv:1910.13797. 
  Joint with Richard Aoun and Marwa Banna.
• Outliers in spectrum of sparse Wigner matrices, Random Struct Alg. 2020; 1-89, arXiv:1904.07985.
  Joint with Konstantin Tikhomirov.
• The rank of random regular digraphs of constant degree, Journal of Complexity, Vol. 48, 2018, pp.103-110., arXiv:1801.05577. 
  Joint with Anna Lytova, Alexander Litvak, Konstantin Tikhomirov and Nicole Tomczak-Jaegermann.
• Circular law for sparse random regular digraphs, J. Eur. Math. Soc. 23, 467-501 (2021), arXiv:1801.05576.
  Joint with Anna Lytova, Alexander Litvak, Konstantin Tikhomirov and Nicole Tomczak-Jaegermann. 
• Structure of eigenvectors of random regular digraphs, Trans. Amer. Math. Soc., 371 (2019), 8097-8172, arXiv:1801.05575.
  Joint with Anna Lytova, Alexander Litvak, Konstantin Tikhomirov and Nicole Tomczak-Jaegermann.
• The dimension-free structure of nonhomogeneous random matrices, Inventiones mathematicae (2018), Volume 214, No. 3, pp. 1031-1080, arXiv:1711.00807.
  Joint with Rafal Latala  and Ramon Van Handel.
  Expository note (in French): CNRS Actualité Scientifique, May 2019.
• The smallest singular value of a shifted d-regular random square matrix, Prob. Th. Rel. Fields, 173 (2019), 1301–1347, arXiv:1707.02635.
  Joint with Anna Lytova, Alexander Litvak, Konstantin Tikhomirov and Nicole Tomczak-Jaegermann.
• The spectral gap for dense random regular graphs, Ann. Probab. 47 (2019), no. 1, 362–419, arXiv:1610.01765.
  Joint with Konstantin Tikhomirov.
• On the norm of a random jointly exchangeable matrix, Journal of Theoretical Probability (2018), arXiv:1610.01751.
  Joint with Konstantin Tikhomirov.
• Approximating matrices and convex bodies, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–19,  arXiv:1605.03861.
  Joint with Omer Friedland.
• Restricted invertibility revisited, A Journey Through Discrete Mathematics. A Tribute to Jiri Matousek., Springer International Publishing, 657–691, 2017, arXiv:1601.00948.
  Joint with Assaf Naor.
• Adjacency matrices of random digraphs: singularity and anti-concentration, J. of Math. Analysis and Appl., 445 (2017), 1447–1491, arXiv:1511.00113.
  Joint with Anna Lytova, Alexander Litvak, Konstantin Tikhomirov and Nicole Tomczak-Jaegermann.
  Winner of 2017 JMAA Ames Award.
• Anti-concentration property for random digraphs and invertibility of their adjacency matrices, C.R. Math. Acad. Sci. Paris, 354 (2016), 121–124. pdf.
  Joint with Anna Lytova, Alexander Litvak, Konstantin Tikhomirov and Nicole Tomczak-Jaegermann.
• Bernstein type inequality for a class of dependent random matrices,  Random Matrices Theory Appl. 5 (2016), no. 2, 1650006, 28 pp, arXiv:1504.05834.
  Joint with Marwa Banna and Florence Merlevède.
• When does a discrete-time random walk in R^n absorb the origin into its convex hull?, Ann. Probab. 45 (2017), no. 2, 965-1002, arXiv:1410.0458v2.
  Joint with Konstantin Tikhomirov.
• Extracting a basis with fixed block inside a matrix,  Linear Algebra Appl. 469 (2015), 28-38, arXiv:1401.6434.
• Estimating the covariance of random matrices, Electron. J. Probab. 18 (2013), no. 107, 26 pp, arXiv:1301.6607.
• A note on column subset selection,  Int. Math. Res. Not. IMRN 2014, no. 23, 6431–6447, arXiv:1212.0976.
• Restricted invertibility and the Banach-Mazur distance to the Cube,  Mathematika 60 (2014), no. 1, 201–218, arXiv:1206.0654.
      
  

Dissertations and expository notes

• Matrices to random taste: phenomena in high dimension, Habilitation thesis. 
• Bennett-Bernstein inequality and the spectral gap of random regular graphs, 9 hours mini-course given at the Lebanese University, Faculty of Sciences II.
• Invertibilité restreinte, distance au cube et covariance de matrices aléatoires, PhD thesis under the supervision of Olivier Guédon.
• Invertibilité restreinte et distance de Banach-Mazur au Cube, Master thesis under the supervision of Dario Cordero-Erausquin et Olivier Guédon. 
• Existence et unicité des préduaux des espaces de Banach, Research project under the supervision of Gilles Godefroy.