Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany.
Centre for Networks and Collective Behaviour, Department of Mathematical Sciences, University of Bath, Bath BA27AY, United Kingdom.
Phys Rev E. 2019 Jul;100(1-1):010302. doi: 10.1103/PhysRevE.100.010302.
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between k tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.
著名的椭圆定律描述了具有非对角元素之间相关性的随机矩阵的特征值分布，该定律在广泛的物理和生物系统中都有应用。在这里，我们研究了将该定律推广到具有 k 个矩阵元素元组之间更高阶循环相关性的随机矩阵的情况。我们表明，该集合中的特征值谱被一个具有 k 重旋转对称性的拟摆线曲线所限制。这个拟摆线定律适用于全矩阵和稀疏矩阵，因此具有显著的普遍性。我们进一步将我们的分析扩展到具有竞争循环模式的矩阵和图，它们更一般地由多摆线谱边界来描述。
