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14
Products of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Rel
 Fields
, 2005
"... ABSTRACT. We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for ..."
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Cited by 30 (1 self)
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ABSTRACT. We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues. 1. INTRODUCTION. In this paper, we consider the asymptotic distribution of eigenvalues of a random matrix of the form πn ˜πnπn where πn and ˜πn are independent n×n random orthogonal projections, of ranks qn and ˜qn, whose distributions are invariant under unitary conjugation. This question is part of a more general
Asymptotic freeness almost everywhere for random matrices
 Acta Sci. Math. (Szeged
, 2000
"... Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather ..."
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Cited by 18 (0 self)
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Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gaussian matrices) by applying the first result to the unitary parts of their diagonalization. Biunitarily invariant nonselfadjoint random matrices are also treated via polar decomposition.
Traces in Twodimensional QCD: The LargeN Limit, to appear
 in Traces in Geometry, Number Theory, and Quantum
"... Abstract. An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the largeN limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops. 1. ..."
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Cited by 11 (3 self)
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Abstract. An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the largeN limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops. 1.
2005): A matrix representation of the BercoviciPata bijection
 Electron. J. Probab
"... E l e c t r o n ..."
FREE PROBABILITY FOR PROBABILISTS
, 1998
"... This is an introduction to some of the most probabilistic aspects of free probability theory. ..."
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Cited by 6 (1 self)
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This is an introduction to some of the most probabilistic aspects of free probability theory.
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.
Free analysis questions II: The Grassmannian completion and the series expansions at the origin, preprint
"... Abstract. The fully matricial generalization in part I, of the difference quotient derivation on holomorphic functions, in which C is replaced by a Banach algebra B, is extended from the affine case to a Grassmannian completion. The infinitesimal bialgebra duality, the duality transform generalizing ..."
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Cited by 5 (0 self)
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Abstract. The fully matricial generalization in part I, of the difference quotient derivation on holomorphic functions, in which C is replaced by a Banach algebra B, is extended from the affine case to a Grassmannian completion. The infinitesimal bialgebra duality, the duality transform generalizing the Stieltjes transform and the spectral theory with noncommuting scalars all extend to this completion. The series expansions of fully matricial analytic functions are characterized, providing a new way to generate fully matricial functions. 1.
Combinatorial aspects of Connes’s embedding conjecture and asymptotic distribution of traces of products of unitaries, Operator Theory 20
 Theta Ser. Adv. Math
, 2006
"... ABSTRACT. In this paper we study the asymptotic distribution of the moments of (nonnormalized) traces Tr(w1),Tr(w2),...,Tr(wr), where w1, w2,..., wr are reduced words in unitaries in the group U(N). We prove that as N → ∞ these variables are distributed as normal gaussian variables √ j1Z1,..., √ Z ..."
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Cited by 4 (0 self)
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ABSTRACT. In this paper we study the asymptotic distribution of the moments of (nonnormalized) traces Tr(w1),Tr(w2),...,Tr(wr), where w1, w2,..., wr are reduced words in unitaries in the group U(N). We prove that as N → ∞ these variables are distributed as normal gaussian variables √ j1Z1,..., √ Zr, where j1,..., jr are the number of cyclic rotations of the words w1,..., ws leaving them invariant. This extends a previous result by Diaconis ([4]), where this it was proved, that Tr(U), Tr(U 2),..., Tr(U p) are asymptotically distributed as Z1, √ 2Z2,..., √ pZp. We establish a combinatorial formula for ∫  Tr(w1)  2 · · · Tr(wp)  2. In our computation we reprove some results from [1]. 1.
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Cited by 4 (1 self)
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
The LargeN YangMills Field on the Plane and Free Noise
"... Abstract. The largeN limit of the quantum YangMills field over the plane R 2, for the gauge group U(N), is shown to lead to a white noise field in the sense of free probability. ..."
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Cited by 2 (1 self)
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Abstract. The largeN limit of the quantum YangMills field over the plane R 2, for the gauge group U(N), is shown to lead to a white noise field in the sense of free probability.