Concentration and convergence rates for spectral measures of random matrices

Meckes, Elizabeth; Meckes, Mark
June 2013
Probability Theory & Related Fields;Jun2013, Vol. 156 Issue 1/2, p145
Academic Journal
The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.


Related Articles

  • Support convergence in the single ring theorem. Guionnet, Alice; Zeitouni, Ofer // Probability Theory & Related Fields;Dec2012, Vol. 154 Issue 3/4, p661 

    We study the eigenvalues of non-normal square matrices of the form A = U T V with U, V independent Haar distributed on the unitary group and T real diagonal. We show that when the empirical measure of the eigenvalues of T converges, and T satisfies some technical conditions, all these...

  • Stochastic aspects of easy quantum groups. Banica, Teodor; Curran, Stephen; Speicher, Roland // Probability Theory & Related Fields;Apr2011, Vol. 149 Issue 3/4, p435 

    We consider several orthogonal quantum groups satisfying the 'easiness' assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr( u) with respect to the Haar measure, u being the fundamental representation. For the classical groups O,...

  • RATES OF CONVERGENCE FOR PRODUCTS OF RANDOM STOCHASTIC 2 x 2 MATRICES. Neininger, Ralph // Journal of Applied Probability;Sep2001, Vol. 38 Issue 3, p799 

    Presents a study which investigated the rates of convergence for products of random stochastic 2 x 2 matrices. Results; Applications to interval splitting; Proofs of theorems.

  • Learnability, Stability and Uniform Convergence. Shalev-Shwartz, Shai; Shamir, Ohad; Srebro, Nathan; Sridharan, Karthik // Journal of Machine Learning Research;10/1/2010, Vol. 11 Issue 10, p2635 

    No abstract available.

  • On the concentration of random multilinear forms and the universality of random block matrices. Nguyen, Hoi; O'Rourke, Sean // Probability Theory & Related Fields;Jun2015, Vol. 162 Issue 1/2, p97 

    The circular law asserts that if $${\mathbf {X}}_n$$ is a $$n \times n$$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of $$\frac{1}{\sqrt{n}} {\mathbf {X}}_n$$ converges almost surely to the uniform distribution on the unit disk as...

  • A connection between quantum dynamics and approximation of Markov diffusions. Morato, L. M.; Ugolini, S. // Journal of Mathematical Physics;Sep94, Vol. 35 Issue 9, p4505 

    It is observed that the existence of an attracting set in the class of solutions to the stochastic Lagrangian variational principle leads to a natural problem of convergence of diffusions in the Carlen class. It is then shown how dynamical properties enable one to prove some convergence results...

  • THE ${L}^{2} $-SINGULAR DICHOTOMY FOR EXCEPTIONAL LIE GROUPS AND ALGEBRAS. HARE, K. E.; JOHNSTONE, D. L.; SHI, F.; YEUNG, W.-K. // Journal of the Australian Mathematical Society;Dec2013, Vol. 95 Issue 3, p362 

    We show that every orbital measure, ${\mu }_{x} $, on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure...

  • An Iterative Solution of Two-Dimensional Birth and Death Processes. Brandwajn, Alexandre // Operations Research;May/Jun79, Vol. 27 Issue 3, p595 

    This paper presents an iterative, seminumerical method for solving the balance equations of finite two-dimensional birth and death processes The method is seminumerical in that it uses the formal knowledge of the stationary probability distribution of one variable, and the iteration is applied...

  • Effect of Ensemble Size on the Spectra of the Deformed Random Matrix Ensemble. Abul-Magd, Ashraf A.; Abul-Magd, Adel Y.; Hassaan, M. Y. // Nature & Science;Apr2013, Vol. 11 Issue 4, p133 

    Random matrix theory (RMT) models the Hamiltonian of a chaotic system by an ensemble of Ndimensional random matrices, where N→∞, conditioned by general symmetry constraints. Some models are introduced to apply RMT to mixed systems. In this paper, we use the deformed random matrices...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics