Statistical properties of structured random matrices.
Statistical properties of structured random matrices.
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结构化随机矩阵的统计特性。
作者：
E. Bogomolny;O. Giraud
研究了元素独立同分布的Hermitian Toeplitz、Hankel和Toeplitz + Hankel随机矩阵的谱性质。结合数值和分析参数，证明了所有这些低复杂度随机矩阵的谱统计是中间类型的，其特征在于：（i）短距离处的能级排斥，（ii）长距离处最近邻分布的指数减小，（iii）谱压缩性的非平凡值，（iv）Fourier空间中特征向量的非平凡分形维数的存在性。我们的研究结果表明，中间型统计量是更普遍和普遍的比被认为是迄今为止，并打开了一个新的方向，随机矩阵理论。
Spectral properties of Hermitian Toeplitz, Hankel, and Toeplitz-plus-Hankel random matrices with independent identically distributed entries are investigated. Combining numerical and analytic arguments it is demonstrated that spectral statistics of all these low-complexity random matrices is of the intermediate type, characterized by: (i) level repulsion at short distances, (ii) an exponential decrease in the nearest-neighbor distributions at long distances, (iii) a nontrivial value of the spectral compressibility, and (iv) the existence of nontrivial fractal dimensions of eigenvectors in Fourier space. Our findings show that intermediate-type statistics is more ubiquitous and universal than was considered so far and open a new direction in random matrix theory.