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- Title
Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability.
- Authors
Collins, Benoît
- Abstract
We consider integrals on unitary groups Ud of the form ∫ UdUi1 j1···Uiq jqUj′1i′1∗···Uj′q′i′q′∗dU. We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows to rederive an asymptotic expansion as d → ∞. Using this, we rederive and strengthen a result of asymptotic freeness due to Voiculescu. We then study large d asymptotics of matrix-model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular, we give an explicit formula for limd→∞(∂n/∂zn )d−2 log∫Udezd Tr(XUYU∗)dU<INNOPIPE>z=0, assuming that the normalized traces d−1Tr(Xk) and d−1Tr (Yk) converge in the large d limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.
- Subjects
RANDOM variables; POLYNOMIALS; FREE probability theory; OPERATOR algebras; ASYMPTOTES; SCHUR functions; MATHEMATICAL formulas
- Publication
IMRN: International Mathematics Research Notices, 2003, Vol 2003, Issue 17, p953
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1155/S107379280320917X