论文标题
图案的随机矩阵：偏离普遍性
Patterned Random Matrices: deviations from universality
论文作者
论文摘要
我们研究了具有附加结构约束的三个真实对称矩阵的三个集合的级别间距分布，以将独立条目的数量减少到$（n+1）/2 $与$ n（n+1）/2 $相比之下，用于$ n \ n \ times n $的$ n（n+1）/2 $。我们以$ 3 \ times 3 $矩阵的分析得出所有结果，并表明间距分布显示了基于结构约束的一系列行为。发现具有附加零的反向循环矩阵的合奏的间距分布比较大间距的指数慢，而对称循环矩阵的间距具有泊松间距。另一方面，alindromic的对称toeplitz矩阵显示出级别的排斥，但分布与Wigner有显着不同。所有三个合奏的间距的行为清楚地表明了对真实对称矩阵的wigner分布的普遍结果。在大的$ n $案例中，与普遍性的偏差也在继续，我们在数字上研究。
We investigate the level spacing distribution for three ensembles of real symmetric matrices having additional structural constraint to reduce the number of independent entries to only $ (n+1)/2 $ in contrast to the $ n(n+1)/2 $ for a real symmetric matrix of size $ n \times n $. We derive all the results analytically exactly for the $ 3\times 3 $ matrices and show that spacing distribution display a range of behaviour based on the structural constraint. The spacing distribution of the ensemble of reverse circulant matrices with additional zeros is found to fall slower than exponential for larger spacing while that of symmetric circulant matrices has poisson spacings. The palindromic symmetric toeplitz matrices on the other hand show level repulsion but the distribution is significantly different from Wigner. The behaviour of spacings for all the three ensembles clearly show the departure from universal result of Wigner distribution for real symmetric matrices. The deviation from universality continues in large $ n $ cases as well, which we study numerically.
