New scaling of Itzykson–Zuber integrals

Abstract

ABSTRACT. We study asymptotics of the Itzykson-Zuber integrals in the scaling when one of the matrices has a small rank compared to the full rank. We show that the result is basically the same as in the case when one of the matrices has a fixed rank. In this way we extend the recent results of Guionnet and Maïda who showed that for a latter scaling the Itzykson-Zuber integral is given in terms of the Voiculescu's Rtransform of the full rank matrix.

The large-Nlimit of matrix integrals over the orthogonal group

“…This was later proved in a rigorous way in [21]. In [18], this is extended to the case where the rank is o(N). Indeed for finite rank of A, say, only terms with a single trace of some power of A dominate, and the expression of F (A, B) is known to be given by n≥1 1 n 1 N Tr A n ψ n (B) for the unitary group [7].…”

Dynamical functional theory for compressed sensing

“…Here, we note that Z({l(t) = 0}) = 1 and therefore direct averaging of Z({l(t)}) is a proper operation. By making use of the R-transform formulation of the asymptotic Itzykson-Zuber integral [23] and the saddle-point method [24] we obtain that Z({l(t)}) factorizes (or decouples) as…”

Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and free probability theory

“…Collins and Sniady [12] have further extended Guionnet and Maïda's result to the case where rank E is o(N ). The same limit as that given by Guionnet and Maïda has been calculated previously by Itzykson and Zuber [13] for β = 2, whose result was extended to the case β = 1 by Marinari, Parisi and Ritort [14].…”