论文标题
$ b $的Freeness和随机矩阵有条件的freeness
Freeness of type $B$ and conditional freeness for random matrices
论文作者
论文摘要
独立单位不变的$ n \ times n $随机矩阵的渐近freens的期望为$ o（n^{ - 2}）$。一个已知的后果是预期的无穷小。我们证明了单位不变的随机矩阵的另一个后果：几乎可以肯定的$ b $的渐近freeness。作为副产品，我们恢复了渐近环状单调性，并获得了渐近条件的烦恼。特别是，两个随机旋转随机矩阵的总和的特征向量经验光谱分布会收敛于有条件的自由卷积。我们还展示了无穷小的Freeness，$ b $的Freeness之间的新联系，有条件的Freeness，循环单调性和单调独立性。最后，我们严格地表明，对于gue矩阵的添加级别对扰动的BBP相变是渐近条件性弗雷格斯的结果，并且参数扩展到对其他单位不变的同时组合的异常值的研究。
The asymptotic freeness of independent unitarily invariant $N\times N$ random matrices holds in expectation up to $O(N^{-2})$. An already known consequence is the infinitesimal freeness in expectation. We put in evidence another consequence for unitarily invariant random matrices: the almost sure asymptotic freeness of type $B$. As byproducts, we recover the asymptotic cyclic monotonicity, and we get the asymptotic conditional freeness. In particular, the eigenvector empirical spectral distribution of the sum of two randomly rotated random matrices converges towards the conditionally free convolution. We also show new connections between infinitesimal freeness, freeness of type $B$, conditional freeness, cyclic monotonicity and monotone independence. Finally, we show rigorously that the BBP phase transition for an additive rank-one perturbation of a GUE matrix is a consequence of the asymptotic conditional freeness, and the arguments extend to the study of the outlier eigenvalues of other unitarily invariant ensembles.
