Hiai, Fumio; Petz, Dénes A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices. (English) Zbl 0954.60030 Ann. Inst. Henri Poincaré, Probab. Stat. 36, No. 1, 71-85 (2000). It is shown that the empirical eigenvalue distribution of suitably distributed random matrices satisfies the large deviation principle as the matrix size goes to infinity. The principal term of the rate function is the logarithmic energy [see P. Koosis, “Introduction to \(H_p\) spaces” (1980; Zbl 0435.30001)] or the minus sign of Voiculescu’s free energy [D. Voiculescu, Commun. Math. Phys. 155, No. 1, 71-92 (1993; Zbl 0781.60006)]. This function is the rate function in the large deviation theorem by G. Ben Arous and A. Guionnet ( see the paper reviewed above), which concerns the empirical distribution of Gaussian symmetric random matrices. Examples of random unitaries are discussed, one of them is related to the work of Gross and Witten in quantum physics. Reviewer: Birute Kryžienė (Vilnius) Cited in 15 Documents MSC: 60F10 Large deviations Keywords:large deviations; random unitary matrix; eigenvalue density; logarithmic energy Citations:Zbl 0435.30001; Zbl 0781.60006; Zbl 0954.60029 PDFBibTeX XMLCite \textit{F. Hiai} and \textit{D. Petz}, Ann. Inst. Henri Poincaré, Probab. Stat. 36, No. 1, 71--85 (2000; Zbl 0954.60030) Full Text: DOI Numdam EuDML