Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability

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dc.creatorCollins, Benoit
dc.date2002-05-07
dc.date2002-06-06
dc.date.accessioned2026-07-07T04:29:11Z
dc.date.available2026-07-07T04:29:11Z
dc.descriptionWe consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen 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 $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.
dc.description34 pages, 1 figure
dc.identifierhttps://arxiv.org/abs/math-ph/0205010
dc.identifierhttp://arxiv.org/abs/math-ph/0205010
dc.identifierInt. Math. Res. Not., (17):953-982, 2003.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/57060
dc.subjectMathematical Physics
dc.subjectProbability
dc.titleMoments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
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