Burda Z, Janik R A, Waclaw B
Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Apr;81(4 Pt 1):041132. doi: 10.1103/PhysRevE.81.041132. Epub 2010 Apr 27.
We show that the eigenvalue density of a product X=X1X2...XM of M independent NxN Gaussian random matrices in the limit N-->infinity is rotationally symmetric in the complex plane and is given by a simple expression rho(z,z)=1/Mpisigma(-2/M)|z|(-2+(2/M)) for |z|<or=sigma, and is zero for |z|>sigma. The parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the Gaussian fluctuations. This form of the eigenvalue density is highly universal. It is identical for products of Gaussian Hermitian, non-Hermitian, and real or complex random matrices. It does not change even if the matrices in the product are taken from different Gaussian ensembles. We present a self-contained derivation of this result using a planar diagrammatic technique. Additionally, we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill Pastur-Lindeberg's condition. We provide a numerical evidence supporting this conjecture.
我们证明，在(N\to\infty)的极限情况下，(M)个独立的(N\times N)高斯随机矩阵的乘积(X = X_1X_2\cdots X_M)的特征值密度在复平面上是旋转对称的，并且对于(|z|\leq\sigma)，由一个简单的表达式(\rho(z,\bar{z})=\frac{1}{M\pi\sigma^{-\frac{2}{M}}}|z|^{-\left(2 - \frac{2}{M}\right)})给出，对于(|z|>\sigma)则为零。参数(\sigma)对应于圆形支撑的半径，并且与高斯涨落的幅度有关。这种特征值密度的形式具有高度的普遍性。它对于高斯厄米特矩阵、非厄米特矩阵以及实或复随机矩阵的乘积是相同的。即使乘积中的矩阵取自不同的高斯系综，它也不会改变。我们使用平面图解技术给出了这个结果的自包含推导。此外，我们推测这种分布对于任何其元素是具有有限方差的独立中心随机变量的矩阵，甚至更一般地对于满足帕斯特 - 林德伯格条件的矩阵也成立。我们提供了支持这个推测的数值证据。
