Abu Dhabi Stochastics Days

Abstract: Let W be a symmetric matrix with i.i.d centered entries with unit variance. The celebrated Wigner’s theorem states that the empirical law of eigenvalues of W, properly normalized, converges weakly to the semicircular law. In this talk, we explore the universality and stability of Wigner’s semicircular law under sparsification and variance profiles. In other words, we consider a random matrix X= Σ∘W, where  Σ is a deterministic matrix and ∘ is the Hadamard product. Among several results, we prove necessary and sufficient conditions for the universality of the semicircular law and the existence of outliers in the spectral distribution. This is a joint work with Dylan Altschuler, Konstantin Tikhomirov, and Pierre Youssef.

The prototypical example of this is the continuum (scaling) limit of the two-dimensional critical Ising model. The case of critical percolation is more difficult, partly because its continuum limit is believed to be described by a relatively unusual type of CFT, called a logarithmic CFT. In this talk, I will first briefly explain the statements above. I will then present some recent results and work in progress that are part of a program aimed at fitting percolation within the logarithmic CFT framework. 

Abstract: In this talk, we will delve into the asymptotic study of simple linear generative models when both the sample size and data dimension grow to infinity. In this high-dimensional regime, random matrix theory (RMT) appears to be a natural tool to assess the model’s performance by examining its asymptotic learned conditional probabilities, its associated fluctuations, and the model’s generalization error. This analytical approach not only enhances our comprehension of generative language models but might also offer novel insights into their refinement through the lens of high-dimensional statistics and RMT.

Abstract: Functional inequalities in discrete settings play a key role in establishing concentration inequalities as well as capturing mixing properties of the underlying dynamics. We investigate the hierarchy of some standard functional inequalities, namely log-Sobolev inequality (LSI), modified log-Sobolev inequality (MLSI), and Poincaré inequality (PI). Using a regularization trick, we provide sharp general comparisons between those inequalities. We present several applications of this, implementing general comparison procedures for Markov chains as well as answering several open problems regarding mixing times. We also investigate connections to discrete curvature and discrete transport inequalities. This is based on joint works with Justin Salez and Konstantin Tikhomirov.