The distribution of eigenvalues of randomized permutation matrices
The distribution of eigenvalues of randomized permutation matrices
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随机排列矩阵的特征值分布
DOI：
10.5802/aif.2777
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
A. Nikeghbali
中科院分区：
文献类型：
--
作者：
J. Najnudel;A. Nikeghbali
在本文中，我们详细研究了一系列随机矩阵系综，这些系综是通过用更一般的非零复数随机变量替换等于 1 的条目，从随机排列矩阵（根据参数 $\theta>0$ 的 Ewens 测度随机选择）获得的。对于这些系综，与高斯酉系综或圆酉系综等更经典的模型相比，可以通过使用排列的循环结构非常明确地计算特征值。此外，通过使用所谓的虚拟排列（由 Kerov、Olshanski 和 Vershik 首先提出，并由 Tsilevich 从概率角度进行研究），我们能够在相同的概率空间上为每个大于或等于 1 的维度定义一个模型，这为维度趋于无穷大时几乎肯定收敛的概念赋予了意义。在本文中，根据所考虑的精确模型，我们获得了特征值点测度的许多不同的收敛结果，其中一些结果给出了很强的收敛性，这在随机矩阵理论中并不常见。
In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $\theta>0$) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations. Moreover, by using the so-called virtual permutations, first introduced by Kerov, Olshanski and Vershik, and studied with a probabilistic point of view by Tsilevich, we are able to define, on the same probability space, a model for each dimension greater than or equal to one, which gives a meaning to the notion of almost sure convergence when the dimension tends to infinity. In the present paper, depending on the precise model which is considered, we obtain a number of different results of convergence for the point measure of the eigenvalues, some of these results giving a strong convergence, which is not common in random matrix theory.