Secular determinants of random unitary matrices
Secular determinants of random unitary matrices
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DOI：
10.1088/0305-4470/29/13/029
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发表时间：
1996-07-07
期刊：
影响因子：
--
通讯作者：
Zyczkowski, K
中科院分区：
文献类型：
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作者：
Haake, F;Kus, M;Zyczkowski, K
我们考虑了由各种循环系综所产生的随机酉矩阵U的特征多项式。特别是，这些多项式的系数的统计进行了研究。这些“长期系数”的方差明确给出任意尺寸和继续分析任意值的水平排斥指数β。后者的长期系数与牛顿著名公式的U的幂的迹有关。虽然迹线倾向于具有高斯分布，并且随着矩阵维度变大而在极限中彼此之间在统计上独立，但是由于迹线与系数的牛顿混合，长期系数表现出强的相互关联。这些结果可能成为相关的，目前的努力结合半经典和随机矩阵理论的量子治疗经典混沌动力学。
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.