Strong asymptotic freeness for Wigner and Wishart matrices preprint

By M. Capitaine, Catherine Donati-Martin. 2005

70 citations recorded by the source corpus.

For each n in N, let X n = [(X n ) jk ] n j,k=1 be a random Hermitian matrix such that the n 2 random variables √n(X n ) ii , √2nRe((X n ) ij ) i<j , √2n Im((X n ) ij ) i<j are independent, identically distributed, with common distribution μ on R. Let X (1) n ,..., x (r) n be r independent copies of X n and (x 1 ,...,x r ) be a semicircular system in a C*-probability space with a faithful state. Assuming that p is symmetric and satisfies a Poincare inequality, we show that, almost everywhere, for any non commutative polynomial p in r variables, (0.1) lim n→∞ ∥p(X (1) n ,..., X (r) n )∥ = ∥p(x 1 ,..., x r )∥. We follow the method of [10] and [17] which gave (0.1) in the Gaussian (complex, real or symplectic) case. We also get that (0.1) remains true when the X (i) n are Wishart matrices while the x i are Marchenko-Pastur distributed.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2e47b6f755559b80c2dbb0493751f9b5c7f2b8e8

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Citations in the corpus

Free Probability and Random Matrices (2014, background): “So we are, for example, not addressing such exciting topics as free stochastic and Malliavin calculus [40, 108, 114], or the rectangular version of free probability [28], or the strong version of asymptotic freeness [48, 58, 89], or free monotone transport [83], or the relation with representation t”
The norm of polynomials in large random and deterministic matrices (2010, background): “Furthermore, let mention that it is clear that we always can take one of the vi to be zero.”
The singular values and vectors of low rank perturbations of large rectangular random matrices (2011, background): “Under additional hypotheses on the manner in which the empirical singular distribution of X n a.s. −→ µ X as n −→ ∞ , Theorem 2.10 can be generalized to any singular value with limit b such that D ′ µ X ( ρ ) is inﬁnite.”
Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices (2010, unknown): “F´evrier ¶”
The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. (2007, methodology): “d just to point out some diﬀerences with the real model case (at the end of Section 4.3) since the approach would be basically the same. In these sections, we will often refer the reader to the paper [13] where the authors deal with several independent non-Deformed Wigner matrices. The reader needs ”