Ja n 20 02 Recursion relations for Unitary integrals , Combinatorics and the Toeplitz Lattice
نویسندگان
چکیده
Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. Email: [email protected]. The support of a National Science Foundation grant # DMS-01-00782 is gratefully acknowledged. Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. E-mail: [email protected] and @math.brandeis.edu. The support of a National Science Foundation grant # DMS-01-00782, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged.
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m at h - ph / 0 20 90 19 v 2 2 4 Se p 20 02 On some integrals over the U ( N ) unitary group and their large N limit
The integral over the U (N) unitary group I = DU exp TrAU BU † is reexamined. Various approaches and extensions are first reviewed. The second half of the paper deals with more recent developments: relation with integrable Toda lattice hierarchy, diagrammatic expansion and combinatorics, and what they teach us on the large N limit of log I.
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The integral over the U (N) unitary group I = DU exp TrAU BU † is reexamined. Various approaches and extensions are first reviewed. The second half of the paper deals with more recent developments: relation with integrable Toda lattice hierarchy, diagrammatic expansion and combinatorics, and what they teach us on the large N limit of log I.
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With the change of variables U = U*q, substitution into (21) results in UCU*q = Ud. Consequently, the solution vector U of (20) can be obtained from the solution vector q of (22) or q can be obtained from U. Note that if a system of real equations is Toeplitz-plus-Hankel (T + H) , where T i s symmetric Toeplitz and H i s skew-centrosym-metric Hankel, then the equations may be transformed into H...
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