Problem of study in secondary schools

Thesis (Ed.M.)--Boston Universit

Large deviations for near-extreme eigenvalues in the beta-ensembles

For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.Comment: We fixed typos and changed Remarks 2.13 and 2.1

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

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

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 $$W^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2$$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.Comment: Random matrices: Theory and Applications (RMTA) To appear (2012) http://www.editorialmanager.com/rmta

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