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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 (9 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
Classical and free infinitely divisible distributions and random matrices
, 2008
"... We construct a random matrix model for the bijection Ψ between classical and free infinitely divisible distributions: for every d ≥ 1, we associate in a quite natural way to each ∗infinitely divisible distribution µ a distribution P µ d on the space of d×d hermitian matrices such that P µ d ∗ Pνd = ..."
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Cited by 9 (4 self)
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We construct a random matrix model for the bijection Ψ between classical and free infinitely divisible distributions: for every d ≥ 1, we associate in a quite natural way to each ∗infinitely divisible distribution µ a distribution P µ d on the space of d×d hermitian matrices such that P µ d ∗ Pνd = Pµ∗ν d. The spectral distribution of a random matrix with distribution P µ d converges in probability to Ψ(µ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the MarchenkoPastur distribution. In an analogous way, for every d ≥ 1, we associate to each ∗infinitely divisible distribution µ a distribution L µ d on the space of complex (nonhermitian) d×d random matrices. If µ is symmetric, the symmetrization of the spectral distribution of Md, when Md is L µ ddistributed, converges
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.
ASYMPTOTICALLY LIBERATING SEQUENCES OF RANDOM UNITARY MATRICES
"... Abstract. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haardistributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices th ..."
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Abstract. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haardistributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation ” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the secondnamed author concerning the limiting distribution of singular values of a randomly chosen submatrix of a discrete Fourier transform matrix. 1.
Eigenvalue distributions of sums and products of . ..
, 2005
"... This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.’s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using R ..."
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This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.’s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using R and Stransforms in free probability theory. We also give a direct derivation of the a.e.d. of the sum of certain random matrices which are not free. This is used to determine the asymptotic signaltointerferenceratio of a multiuser CDMA system with a minimum meansquare error linear receiver.