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44
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Stable laws and domains of attraction in free probability theory, With an appendix by Philippe Biane
 Ann. of Math
, 1999
"... with an appendix by Philippe Biane In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws ..."
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Cited by 27 (0 self)
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with an appendix by Philippe Biane In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated. 1.
Commutators of Free Random Variables
 Duke Math. J
, 1998
"... Let A be a unital C algebra, given together with a specied state ' : A ! C. ..."
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Cited by 24 (5 self)
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Let A be a unital C algebra, given together with a specied state ' : A ! C.
On quantum statistical inference
 J. Roy. Statist. Soc. B
, 2001
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section ..."
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Cited by 24 (5 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research Section
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...
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices
 Advances in Mathematics 227 (2011
"... Abstract. In this paper, we consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and ..."
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Cited by 15 (4 self)
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Abstract. In this paper, we consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting nonrandom value is shown to depend explicitly on the limiting spectral measure and the assumed perturbation model via integral transforms that correspond to very well known objects in free probability theory that linearize noncommutative free additive and multiplicative convolution. Moreover, we uncover a remarkable phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. This critical threshold is intimately related to the same aforementioned integral transforms. We examine the consequence of this eigenvalue phase transition on the associated eigenvectors and generalize our results to examine the singular values and vectors of finite, low rank perturbations of rectangular random matrices. The analysis brings into sharp focus the analogous connection with rectangular free probability. Various extensions of our results are discussed. 1.
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
Large Deviations Upper Bounds and Non Commutative Entropies for Some Matrices Ensembles
, 2000
"... ..."
Conditional moments of qMeixner processes
, 2004
"... Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these proce ..."
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Cited by 9 (5 self)
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Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the noncommutative generalizations of the Lévy processes. 1.