Random truncations of Haar distributed matrices and bridges

Let $U$ be a Haar distributed matrix in $mathbb U(n)$ or $mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process [T^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of $U$ by a random one, where each row (resp. column) is chosen with probability $s$ (resp. $t$) independently. We prove that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process. On the way we meet other interesting convergences.

Large deviations for the largest eigenvalue of an Hermitian Brownian motion

We establish a large deviation principle for the process of the largest eigenvalue of an Hermitian Brownian motion. By a contraction principle, we recover the LDP for the largest eigenvalue of a rank one deformation of the GUE.

Truncations of Haar distributed matrices, traces and bivariate Brownian bridges

Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.

Central limit theorems for eigenvalues of deformations of Wigner matrices

In this paper, we explain the dependance of the fluctuations of the largest eigenvalues of a Deformed Wigner model with respect to the eigenvectors of the perturbation matrix. We exhibit quite general situations that will give rise to universality or non universality of the fluctuations.

Further examples of explicit Krein representations of certain subordinators

In a previous paper, we have shown that the gamma subordinators may be represented as inverse local times of certain diffusions. In the present paper, we give such representations for other subordinators whose Levy densities are of the form $ frac{mathcal{C}}{(sinh(y))^gamma}$, $0 < gamma < 2$, and the more general family obtained from those by exponential tilting.

Strong asymptotic freeness for Wigner and Wishart matrices

We prove that any non commutative polynomial of r independent copies of Wigner matrices converges a.s. towards the polynomial of r free semicircular variables in operator norm. This result extends a previous work of Haagerup and Thorbjornsen where GUE matrices are considered, as well as the classical asymptotic freeness for Wigner matrices (i.e. convergence of the moments) proved by Dykema. We also study the Wishart case.

Some explicit Krein representations of certain subordinators, including the Gamma process

We give a representation of the Gamma subordinator as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms. In particular, we have obtained Krein representations of the subordinators which govern the two parameter Poisson-Dirichlet family of distributions.

Large deviations for Wishart processes

Let $X^{(delta)}$ be a Wishart process of dimension $delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes ${delta^{-1} X_t^{(delta)}, t leq 1 }$ as $delta$ tends to infinity. The process $X^{(delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.