论文标题
随机矩阵繁殖的地方法律
Local laws for multiplication of random matrices
论文作者
论文摘要
考虑随机矩阵模型$ a^{1/2} ubu^* a^{1/2}，$，其中$ a $和$ b $是两个$ n \ times n $ n $确定矩阵，$ u $是$ n \ times n $ n $ haar单位或正交随机矩阵。 It is well-known that on the macroscopic scale, the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of $A$ and $B,$ denoted as $μ_α\boxtimes μ_β,$ where $μ_α$ and $μ_β$ are the limiting ESDs of $A$ and $B,$ respectively.在本文中，我们研究了边缘特征值和特征向量统计的渐近显微镜行为。我们证明，$μ_a\ boxtimesμ_b的密度分别为$μ_a$和$μ_b$，分别为$ a $ a $ a $ a $ a $ a $ a和$ b，$，相关的从属功能在边缘附近具有常规行为。此外，我们以最佳规模建立了边缘附近的当地法律。特别是，我们证明分解的条目仅取决于$ a，b $的特征值和最佳收敛速率的下属功能。我们的证明和计算基于[3,5,6,8]中为添加剂模型$ a+ubu^*$开发的技术，我们的结果可以视为乘法模型的[8]的对应物。
Consider the random matrix model $A^{1/2} UBU^* A^{1/2},$ where $A$ and $B$ are two $N \times N$ deterministic matrices and $U$ is either an $N \times N$ Haar unitary or orthogonal random matrix. It is well-known that on the macroscopic scale, the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of $A$ and $B,$ denoted as $μ_α\boxtimes μ_β,$ where $μ_α$ and $μ_β$ are the limiting ESDs of $A$ and $B,$ respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of $μ_A \boxtimes μ_B,$ where $μ_A$ and $μ_B$ are the ESDs of $A$ and $B,$ respectively and the associated subordination functions have a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of $A, B$ and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model $A+UBU^*$ in [3,5,6,8], and our results can be regarded as the counterparts of [8] for the multiplicative model.
