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Spectral measure of large random Hankel, Markov and Toeplitz matrices
 Ann. Probab
"... Abstract. We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of zero mea ..."
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Cited by 23 (8 self)
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Abstract. We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {Xk} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of zero mean and unit variance, scaling the eigenvalues by √ n we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γH, γM, and γT of unbounded support. The moments of γH and γT are the sum of volumes of solids related to Eulerian numbers, whereas γM has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables {Xi,j}j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m = 0, and the fourth moment is finite, we prove that the spectral norm of Mn scaled by √ 2n log n converges almost surely to one. 1. Introduction and
Free Diffusions, Free Entropy And Free Fisher Information
"... . Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free ..."
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Cited by 19 (0 self)
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. Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t ! 1, towards the master field (i.e. the Gibbs state). 1. Introduction The purpose of this paper is to start the study of diffusion equations where the driving noise is a free brownian motion. Reasons for considering such equations will be explained in the next sections of this introduction. 1.1 Gibbs states and diffusion theory. Let V be a C 2 function on R d , with Z = Z R d e \GammaV (x) dx ! 1: The probability measur...
COMPUTATION OF SOME EXAMPLES OF BROWN’S SPECTRAL MEASURE IN FREE PROBABILITY
, 2000
"... We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples includeun+u1where unandu1are the generators of Zn and Z respectively, in the free product ZnZ, or elliptic elements, of the form S+iS where S and S a ..."
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Cited by 17 (0 self)
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We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples includeun+u1where unandu1are the generators of Zn and Z respectively, in the free product ZnZ, or elliptic elements, of the form S+iS where S and S are free semicircular elements of variance and.
Large deviations and stochastic calculus for large random matrices
, 2004
"... Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of math ..."
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Cited by 13 (0 self)
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Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the socalled matrix models or in free probability by the definition of a noncommutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non freeprobabilists.
A free probability analogue of the Wasserstein metric on the tracestate space
 Geom. Funct. Anal
"... Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semicircle distribution is majorized by a modified free entropy quantity. 0 ..."
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Cited by 13 (0 self)
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Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semicircle distribution is majorized by a modified free entropy quantity. 0
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
Moderate Deviations for the Spectral Measure of Certain Random Matrices
 ANN. INST. H. POINCARÉ(B) PROBAB. STATIST
, 2002
"... We derive a moderate deviations principle for matrices of the form XN = DN + WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit D . Our techniques are based on a dynamical approach introduced by CabanalDuvill ..."
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Cited by 5 (1 self)
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We derive a moderate deviations principle for matrices of the form XN = DN + WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit D . Our techniques are based on a dynamical approach introduced by CabanalDuvillard and Guionnet.
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
, 2008
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Limit theorems in free probability theory. I
 I ARXIV:MATH. OA/0602219 V
, 2006
"... Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for nonidentically distributed random variables in classical Probability Theory. ..."
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Cited by 4 (1 self)
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Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for nonidentically distributed random variables in classical Probability Theory.