Exact statistical properties of the zeros of complex random polynomials
Exact statistical properties of the zeros of complex random polynomials
复制标题
DOI：
10.1088/0305-4470/32/16/006
复制
发表时间：
1998-12
期刊：
影响因子：
--
通讯作者：
P. Forrester;G. Honner
中科院分区：
文献类型：
--
作者：
P. Forrester;G. Honner
复高斯随机多项式的零点，其系数使得底层复空间中的密度是均匀的，已知具有与一维混沌量子系统的相干态表示的零点相同的统计性质。我们扩展了这些多项式的解释，表明它们也出现在磁场中的量子粒子的波函数构造从一个随机叠加的状态在最低的朗道水平。零点的统计特性的研究进行使用精确的公式为一点和两点分布函数。注意力集中在两点相关性的大部分，线性统计量的方差的时刻，和两点相关性在边界处的渐近形式。对复高斯随机矩阵的特征值作了相同量的比较。
The zeros of complex Gaussian random polynomials, with coefficients such that the density in the underlying complex space is uniform, are known to have the same statistical properties as the zeros of the coherent state representation of one-dimensional chaotic quantum systems. We extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest Landau level. A study of the statistical properties of the zeros is undertaken using exact formulae for the one- and two-point distribution functions. Attention is focused on the moments of the two-point correlation in the bulk, the variance of a linear statistic, and the asymptotic form of the two-point correlation at the boundary. A comparison is made with the same quantities for the eigenvalues of complex Gaussian random matrices.