Fritz Haake et al 1996 J. Phys. A: Math. Gen. 29 3641 doi:10.1088/0305-4470/29/13/029
Secular determinants of random unitary matrices
Fritz Haake
, Marek Kus,
, Hans-Jürgen Sommers, Henning Schomerus and Karol Zyczkowski,§
We consider the characteristic polynomials of random unitary matrices U drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The variances of these `secular coefficients' are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent 
. The latter secular coefficients are related to the traces of powers of U by Newton's well known formulae. While the traces tend to have Gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large, the secular coefficients exhibit strong mutual correlations due to Newton's mixing of traces to coefficients. These results might become relevant for current efforts at combining semiclassics and random-matrix theory in quantum treatments of classically chaotic dynamics.
- PACS
- MSC
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15A15 Determinants, permanents, other special matrix functions (See also 19B10, 19B14)
- Subjects
- Dates
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Issue 13 (7 July 1996)
Received 19 January 1996
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Secular determinants of random unitary matrices
Fritz Haake et al 1996 J. Phys. A: Math. Gen. 29 3641
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