We derive exact analytical expressions for correlation functions of singular values of the product of $M$ Ginibre matrices of size $N$ in the double scaling limit $M,N\to \infty$. The singular value statistics is described by a determinantal point process with a kernel that interpolates between Gaussian unitary ensemble statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit $N\to \infty$, the interpolation parameter is given by the limiting quotient $a=N/M$. One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk, and at the soft edge for any $a$. We find that in addition to the standard scaling regimes, there is a transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

中文翻译：

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随机矩阵乘积的局部光谱统计的通用性

我们得出以下乘积的奇异值的相关函数的精确解析表达式 $\mathrm{中号}$ Ginibre矩阵的大小 $ñ$ 在双倍缩放限制中 $\mathrm{中号}，ñ\to \infty$。奇异值统计量是通过确定性点过程来描述的，该过程具有在高斯unit合奏统计量和Dirac-delta（小栅栏）统计量之间进行插值的内核。在热力学极限$ñ\to \infty$，插值参数由极限商给出 $\mathrm{一种}=ñ/\mathrm{中号}$。我们的目标之一是在任何地方的硬边缘，大块和软边缘找到显式的内核形式$\mathrm{一种}$。我们发现，除了标准的缩放比例机制外，还有一个过渡机制可以插在硬边和主体之间。我们推测这些结果是普遍的，它们适用于来自高斯吸引盆地的各种随机矩阵产品，包括相关矩阵。我们通过数值模拟证实了这一猜想。此外，我们表明，考虑的随机矩阵乘积的局部频谱统计信息与戴森布朗运动的局部统计信息相同，初始条件由等距位置给出，但关键的区别在于该等价性仅局部存在。最后，我们在软边缘处确定了介观光谱尺度，这对于光谱的展开至关重要。