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Eigenvalue variance bounds for Wigner and covariance random matrices
 RMTA
"... Abstract. This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds ..."
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Abstract. This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices. Two different models of random Hermitian matrices were introduced by Wishart in the twenties and by Wigner in the fifties. Wishart was interested in modeling tables of random data in multivariate analysis and worked on random covariance matrices. In this paper, the results for covariance matrices are very close to those for Wigner matrices. Therefore, it deals mainly with Wigner matrices. Definitions and results regarding covariance matrices are available in the last section.
Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
, 2013
"... The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painleve ́ II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is s ..."
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The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painleve ́ II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painleve ́ II that arises is a double shifted Bäcklund transformation of the Hastings–McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hardtosoft edge transition, involving the limit as the repulsion strength at the hard edge a → ∞, to existing results for the joint distribution of the first and second eigenvalue at the hard edge (Forrester and Witte 2007 Kyushu J. Math. 61 457–526). In addition recursions under a → a + 1 of quantities specifying the latter are obtained. A Fredholm determinant type characterization is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.
ON SOME SPECIAL DIRECTED LASTPASSAGE PERCOLATION MODELS
, 2007
"... Abstract. We investigate extended processes given by lastpassage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to nonuniversal and trivial extended processes. 1. Introduction and ..."
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Abstract. We investigate extended processes given by lastpassage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to nonuniversal and trivial extended processes. 1. Introduction and
Large Complex Correlated Wishart Matrices: Fluctuations and Asymptotic Independence at the Edges.
, 2014
"... We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show t ..."
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We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the TracyWidom law at the scale N2/3. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hardedge TracyWidom law at the scale N2. As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided.
UNIVERSALITY IN UNITARY RANDOM MATRIX ENSEMBLES WHEN THE SOFT EDGE MEETS THE
, 2007
"... Dedicated to Percy Deift on the occasion of his sixtieth birthday Abstract. Unitary random matrix ensembles Z −1 n,N (detM)α exp(−N Tr V (M))dM defined on positive definite matrices M, where α> −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically v ..."
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Dedicated to Percy Deift on the occasion of his sixtieth birthday Abstract. Unitary random matrix ensembles Z −1 n,N (detM)α exp(−N Tr V (M))dM defined on positive definite matrices M, where α> −1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when n, N → ∞ such that n/N −1 = O(n −2/3). For each value of α> −1 we find a oneparameter family of limiting kernels that we describe in terms of the HastingsMcLeod solution of the Painlevé II equation with parameter α + 1/2.