论文标题
随机矩阵的渐近环条件
Asymptotic cyclic-conditional freeness of random matrices
论文作者
论文摘要
Voiculescu的Freeness在计算$ n \ times n $随机矩阵上具有特征位置的多项式光谱的渐近性出现：它们以通用位置为准：它们是用统一的单位随机矩阵$ u_n $随机旋转的。在本文中，我们通过提出一个随机矩阵模型来详细阐述上几点，我们将其命名为Vortex模型，其中$ u_n $具有统一的统一随机矩阵法律，该法律条件旨在留下不变的一个确定性矢量$ v_n $。在限制$ n \ to +\ infty $中，我们表明，矩阵$ u_n $随机旋转的$ n \ times n $矩阵在正常的跟踪和状态向量$ v_n $方面是渐近免费的。为了描述二阶渐近学，我们定义了循环条件的弗雷尼斯，这是一个独立的新概念，它统一了无穷小的弗雷尼斯，环状 - 单酮独立性和环状树树独立性。由于这种新的独立性，可以计算涡流模型中的无限分布。最后，我们详细介绍了Vortex模型，以构建有序的Freeness和缩进独立性的随机矩阵模型。
Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on $N\times N$ random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix $U_N$. In this article we elaborate on the previous point by proposing a random matrix model, which we name the Vortex model, where $U_N$ has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector $v_N$. In the limit $N \to +\infty$, we show that $N\times N$ matrices randomly rotated by the matrix $U_N$ are asymptotically conditionally free with respect to the normalized trace and the state vector $v_N$. To describe second order asymptotics, we define cyclic-conditional freeness, a new notion of independence unifying infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. The infinitesimal distribution in the Vortex model can be computed thanks to this new independence. Finally, we elaborate on the Vortex model in order to build random matrix models for ordered freeness and for indented independence.
