标题： 自由卷积和广义Dyson布朗运动 显示英文标题

标题： Free Convolution and Generalized Dyson Brownian Motion

摘要： 大型随机矩阵之和的特征值谱，当这些矩阵相互“自由”，即随机旋转时，可以使用R变换的形式主义来获得，在不同领域有许多应用。 我们提供了一个直接解释，将R变换的otherwise抽象的可加性性质应用于求和，以“粒子”（特征值）的动力学演化为形式，这些粒子通过两体和高体作用力相互作用，并受到高斯噪声的影响，推广了通常的Dyson布朗运动与库仑相互作用。 有趣的是，谱的主体之外出现异常值由广义Dyson运动的“速度”的发散所标志。 我们将结果扩展到自由矩阵的乘积。 显示英文摘要

摘要： The eigenvalue spectrum of the sum of large random matrices that are mutually "free", i.e., randomly rotated, can be obtained using the formalism of R-transforms, with many applications in different fields. We provide a direct interpretation of the otherwise abstract additivity property of R-transforms for the sum in terms of a dynamical evolution of "particles" (the eigenvalues), interacting through two-body and higher-body forces and subject to a Gaussian noise, generalizing the usual Dyson Brownian motion with Coulomb interaction. Interestingly, the appearance of an outlier outside of the bulk of the spectrum is signalled by a divergence of the "velocity" of the generalized Dyson motion. We extend our result to products of free matrices.