Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.

By Hanne Schultz. 2003

58 citations recorded by the source corpus.

Abstract.In [HT2] Haagerup and Thorbjo rnsen prove the following extension of Voiculescu’s random matrix model (cf. [V2, Theorem 2.2]): For each n ∈ ℕ, let X1(n),..., Xr(n) be r independent complex self-adjoint random matrices from the class and let x1,...,xr be a semicircular system in a C*-probability space. Then for any polynomial p in r non-commuting variables the convergence holds almost surely. We generalize this result to sets of independent Gaussian random matrices with real or symplectic entries (the GOE- and the GSE-ensembles) and random matrix ensembles related to these.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/694178ba06d57e40b8b68e22fdc28c3b21a6f7ec

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Citations in the corpus

Strong asymptotic freeness for Wigner and Wishart matrices preprint (2005, background): “…the remainder of the two proofs (in [9] and [15]) deals with the estimation of lim sup where h is an analytic function on C \ R which satisﬁes In [9] section 6, Haagerup and Thorbjørnsen introduce a very clever family of functions { I p ( λ ) , p ≥ 1 deﬁned by where k is the degree of the…”
Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices (2010, methodology): “…complex, real or symplectic Wigner setting; nevertheless, since µ σ µ A N depends on N , we need here to apply the inverse Stieltjes tranform to functions depending on N and we therefore give the details of the proof to convince the reader that the approach developped by [23] and [31] still holds.”
The norm of polynomials in large random and deterministic matrices (2010, methodology): “Their method has been used by Schultz [ 31 ] to obtain the same result for Gaussian random matrices with real or symplectic entries, and by Capitaine and Donati-Martin [10] for Wigner matrices with symmetric distribution of the entries satisfying a Poincare inequality and for Wishart matrices.”
Convergence of the largest singular value of a polynomial in independent Wigner matrices (2011, methodology): “In this section we introduce the ensemble of tools we will use to replace the Poincaré-type inequalities used in [9], [8], [19], [4] and [14].”