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Medientyp:
E-Artikel
Titel:
Strong Asymptotic Freeness for Wigner and Wishart Matrices
Beteiligte:
Capitaine, M.;
Donati-Martin, C.
Erschienen:
Department of Mathematics of Indiana University, 2007
Erschienen in:Indiana University Mathematics Journal
Sprache:
Englisch
ISSN:
0022-2518;
1943-5258
Entstehung:
Anmerkungen:
Beschreibung:
<p>For each n in ℕ, let ${\mathrm{X}}_{\mathrm{n}}={\left[({\mathrm{X}}_{\mathrm{n}}{)}_{\mathrm{j}\mathrm{k}}\right]}_{\mathrm{j},\mathrm{k}=1}^{\mathrm{n}}$ be a random Hermitian matrix such that the n2 random variables $\sqrt{\mathrm{n}}({\mathrm{X}}_{\mathrm{n}}{)}_{\mathrm{i}\mathrm{i}},\sqrt{2\mathrm{n}}\mathrm{Re}(({\mathrm{X}}_{\mathrm{n}}{)}_{\mathrm{i}\mathrm{j}}{)}_{\mathrm{i}<\mathrm{j}}$, $\sqrt{2\mathrm{n}}\mathrm{I}\mathrm{m}(({\mathrm{X}}_{\mathrm{n}}{)}_{\mathrm{i}\mathrm{j}}{)}_{\mathrm{i}<\mathrm{j}}$ are independent, identically distributed, with common distribution μ on ℝ. Let ${\mathrm{X}}_{\mathrm{n}}^{\left(1\right)},\mathrm{\ldots},{\mathrm{X}}_{\mathrm{n}}^{\left(\mathrm{r}\right)}$ be r independent copies of Xn and (x1,..., xr) be a semicircular system in a C*-probability space with a faithful state. Assuming that μ is symmetric and satisfies a Poincaré inequality, we show that, almost everywhere, for any non commutative polynomial p in r variables, (0.1) $\underset{\mathrm{n}\to +\mathrm{\infty }}{\mathrm{lim}}\Vert \mathrm{p}({\mathrm{X}}_{\mathrm{n}}^{\left(1\right)},\mathrm{\ldots},{\mathrm{X}}_{\mathrm{n}}^{\left(\mathrm{r}\right)})\Vert =\Vert \mathrm{p}({\mathrm{x}}_{1},\mathrm{\ldots},{\mathrm{x}}_{\mathrm{r}})\Vert $. 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 ${\mathrm{X}}_{\mathrm{n}}^{\left(\mathrm{i}\right)}$ are Wishart matrices while the xi are Marchenko-Pastur distributed.</p>