Related papers: Local semicircle law in the bulk for Gaussian $\be…

The Gaussian $\beta$-ensemble (G$\beta$E) is a fundamental model in random matrix theory. In this paper, we provide a comprehensive asymptotic description of the characteristic polynomial of the G$\beta$E anywhere in the bulk of the…

We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the…

The matrix $A:\mathbb{R}^n \to \mathbb{R}^m$ is $(\delta,k)$-regular if for any $k$-sparse vector $x$, $$ \left| \|Ax\|_2^2-\|x\|_2^2\right| \leq \delta \sqrt{k} \|x\|_2^2. $$ We show that if $A$ is $(\delta,k)$-regular for $1 \leq k \leq…

We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the…

We consider the problem of robust mean and location estimation w.r.t. any pseudo-norm of the form $x\in\mathbb{R}^d\to ||x||_S = \sup_{v\in S}<v,x>$ where $S$ is any symmetric subset of $\mathbb{R}^d$. We show that the deviation-optimal…

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only…

We study the hard edge limit of a multilevel extension of the Laguerre $\beta$-ensemble at zero temperature. In particular, we show that asymptotically the ensemble is given by Gaussians with covariance matrix expressible in terms of the…

The sub-optimality of Gauss--Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order $\alpha$, where the optimality is in the sense of worst-case error.…

We study the extreme point process associated to the off-diagonal components in the matrix representation of the Gaussian $\beta$-Ensemble and prove its convergence to Poisson point process as $n\to +\infty$ when the inverse temperature…

We study sampling algorithms for $\beta$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a…

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the…

For any $\beta>0$, we provide a tridiagonal matrix model and compute the joint eigenvalue density of a random rank one non-Hermitian perturbation of Gaussian and Laguerre $\beta$-ensembles of random matrices.

The spectral density for random matrix $\beta$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $\beta$, which for even $\beta$ is a polynomial of degree…

A local optimization method based on Bayesian Gaussian Processes is developed and applied to atomic structures. The method is applied to a variety of systems including molecules, clusters, bulk materials, and molecules at surfaces. The…

We investigate the characteristic polynomials $\varphi_N$ of the Gaussian $\beta$-ensemble for general $\beta>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for $\varphi_N$ in terms of…

We provide optimal upper bounds on the growth of iterated sumsets $hA=A+\dots+A$ for finite subsets $A$ of abelian semigroups. More precisely, we show that the new upper bounds recently derived from Macaulay's theorem in commutative algebra…

We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an…

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the…

The aim of this paper is to provide a novel proof for the Local Semicircle Law for the Wigner ensemble. The core of the proof is the intensive use of the algebraic structure that arises, i.e. resolvent expansions and resolvent identities.…

This paper studies the partial estimation of Gaussian graphical models from high-dimensional empirical observations. We derive a convex formulation for this problem using $\ell_1$-regularized maximum-likelihood estimation, which can be…