Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Secular determinants of random unitary matrices.
AU - Haake, Fritz
AU - Kus, Marek
AU - Sommers, Hans-Jurgen
AU - Schomerus, Henning
AU - Zyczkowski, Karol
PY - 1996
Y1 - 1996
N2 - 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 beta. 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.
AB - 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 beta. 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.
U2 - 10.1088/0305-4470/29/13/029
DO - 10.1088/0305-4470/29/13/029
M3 - Journal article
VL - 29
SP - 3641
JO - Journal of Physics A: Mathematical and General
JF - Journal of Physics A: Mathematical and General
SN - 1361-6447
ER -