A ug 2 00 5 Concentration of Haar measures , with an application to random matrices
نویسنده
چکیده
In this article, we present a general technique for analyzing the concentration of Haar measures on compact groups using the properties of certain kinds of random walks. As an application, we obtain a new kind of measure concentration for random unitary matrices, which allows us to directly establish the concentration of the empirical distribution of eigenvalues of a class of random matrices. The end-result of this application is a quantitative version of Voiculescu's celebrated connection between random matrices and free probability.
منابع مشابه
Concentration of Haar measures , with an application to random matrices
We present a novel approach to measure concentration that works through a deeper investigation of the semigroup method. In particular, we show how couplings and rates of convergence of Markov chains can be used to obtain concentration bounds. As an application, we obtain a measure concentration result for random unitary matrices and other kinds of Haar-distributed random variables, which allows...
متن کاملO ct 2 00 3 On asymptotics of large Haar distributed unitary matrices 1
Entries of a random matrix are random variables but a random matrix is equivalently considered as a probability measure on the set of matrices. A simple example of random matrix has independent identically distributed entries. In this paper random unitary matrices are studied whose entries must be correlated. A unitary matrix U = (Uij) is a matrix with complex entries and UU ∗ = UU = I. In term...
متن کاملDescribing the Behavior of Eigenvectors of Random Matrices Using Sequences of Measures on Orthogonal Groups * Jack
A conjecture has previously been made on the chaotic behavior of the eigenvectors of a class of n-dimensional random matrices, where n is very large [ Evidence supporting the conjecture has been given in the form of two limit theorems, as n-. relating the random matrices to matrices formed from the Haar measure, h,, on the orthogonal group The present paper considers a reformulation of the conj...
متن کاملar X iv : m at h - ph / 0 21 00 58 v 1 3 0 O ct 2 00 2 Random matrix theory , the exceptional Lie groups , and L - functions
Random matrix theory, the exceptional Lie groups, and L-functions. Abstract There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices an...
متن کاملar X iv : 0 81 0 . 27 53 v 1 [ m at h . ST ] 1 5 O ct 2 00 8 Concentration of the spectral measure of large Wishart matrices with dependent entries
We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered.
متن کامل