论文标题
部分可观测时空混沌系统的无模型预测
Universality of the least singular value and singular vector delocalisation for Lévy non-symmetric random matrices
论文作者
论文摘要
在本文中，我们考虑$ n \ times n $矩阵$ d_ {n} $ with I.i.d。所有条目都以下$ a- $稳定的法律除以$ n^{1/a} $。我们证明，乘以$ n $的$ d_ {n} $的最小单数值趋向于与高斯情况相同的法律，几乎是所有$ a \ in（0,2）$。通过考虑矩阵$ d_ {n} $的对称性并使用三步策略的版本，这是随机矩阵理论文献中众所周知的策略，这是通过考虑对称的。为了采用三个步骤策略，我们还证明了一项各向同性局部定律，用于通过具有相似结构的高斯矩阵稍微扰动矩阵的对称性。各向同性的地方法已被证明是满足某些规律性假设的一般矩阵类别。我们还证明，在小能量（即以$ 0 $ $ 0 $的小间隔）以$ d_ {n} $ $ d_ {n} $的左右单数向量的完整离域化。
In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries all following an $a-$stable law divided by $N^{1/a}$. We prove that the least singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the Gaussian case, for almost all $a \in (0,2)$. This is proven by considering the symmetrization of the matrix $D_{N}$ and using a version of the three step strategy, a well known strategy in the random matrix theory literature. In order to apply the three step strategy, we also prove an isotropic local law for the symmetrization of matrices after slightly perturbing them by a Gaussian matrix with a similar structure. The isotropic local law is proven for a general class of matrices that satisfy some regularity assumption. We also prove the complete delocalization for the left and right singular vectors of $D_{N}$ at small energy, i.e., for energies at a small interval around $0$.
