论文标题
结构化随机矩阵的Wigner类型定律
Wigner type laws for structured random matrices
论文作者
论文摘要
对于足够漂亮的2维形状，我们将其近似矩阵（或图案矩阵）定义为随机矩阵，并根据给定模式排列IID条目。对于大近似矩阵，我们观察到特征值大致遵循潜在的分布。这种现象类似于Wigner矩阵上的经典观察。我们证明，随着大小的增加并等于组合定义的函数的积分，该矩阵的矩渐近地收敛，该功能计算有限网格上的某些路径。我们还考虑了几个独立的图案矩阵的情况。在一组特定的条件下，这些矩阵在完整的独立正方形随机矩阵方面接受了渐近的Freeness。在我们的结论中，我们提出了几个开放问题。
For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to a given pattern. For large approximating matrices, we observe that the eigenvalues roughly follow an underlying distribution. This phenomenon is similar to the classical observation on Wigner matrices. We prove that the moments of such matrices converge asymptotically as the size increases and equals to the integral of a combinatorially-defined function which counts certain paths on a finite grid. We also consider the case of several independent patterned matrices. Under a specific set of conditions, these matrices admit asymptotic freeness with respect to full-filled independent square random matrices. In our conclusion, we present several open problems.
