Burda Z, Janik R A, Nowak M A
Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, Reymonta 4, PL-30-059 Kraków, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Dec;84(6 Pt 1):061125. doi: 10.1103/PhysRevE.84.061125. Epub 2011 Dec 14.
We derive a multiplication law for free non-Hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define the corresponding non-Hermitian S transform being a natural generalization of the Voiculescu S transform. In addition, we extend the classical Hermitian S transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered. We use planar diagrammatic techniques to derive these results.
我们推导了自由非厄米随机矩阵的乘法法则，这使得能够根据单个系综的特征轻松重建乘积系综的二维本征值分布。我们定义了相应的非厄米S变换，它是Voiculescu S变换的自然推广。此外，我们扩展了经典的厄米S变换方法，以处理随机矩阵系综因子均值为零的情况，包括两者均为中心分布的情况。我们使用平面图形技术来推导这些结果。
