独立随机高斯矩阵乘积的谱
Spectrum of the Product of Independent Random Gaussian Matrices
摘要：
我们证明了乘积X= X_1X_2.在大N极限下，M个独立的N × N高斯随机矩阵的X_M在复平面上是旋转对称的，当|z| <σ时，X_M由一个简单的表达式rho（z，\bar{z}）= 1/（Mπσ^2} |z| ^{-2+2/M}给出，当|z|> σ时，X_M为零.参数σ对应于圆形支撑的半径，并且与高斯波动的幅度有关。这种形式的特征值密度是高度通用的。它对于高斯厄米特、非厄米特、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。我们提出了一个独立的推导这个结果使用平面图形技术高斯矩阵。我们还给出了一个数值证据表明，这一结果也适用于矩阵的元素是独立的，集中的随机变量与有限的方差。
英文摘要：
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\bar{z}) = 1/(Mπσ^2} |z|^{-2+2/M} for |z|<σ, and is zero for |z|> σ. The parameter σcorresponds 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, 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 for Gaussian matrices. We also give a numerical evidence suggesting that this result applies also to matrices whose elements are independent, centered random variables with a finite variance.
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