Real Second-Order Freeness and the Asymptotic Real Second-Order Freeness of Several Real Matrix Models
Real Second-Order Freeness and the Asymptotic Real Second-Order Freeness of Several Real Matrix Models
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几种实数矩阵模型的实二阶自由度和渐近实二阶自由度
DOI：
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发表时间：
2011
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通讯作者：
C. Redelmeier
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作者：
C. Redelmeier
我们在二阶非交换概率空间中引入实二阶自由度。我们证明，在这个定义下，随机矩阵的三个实数模型，即实数吉尼布矩阵、高斯正交矩阵和实数威沙特矩阵，是渐近二阶自由的。这些系综不满足其复杂类似物所满足的二阶自由度的复杂定义。我们使用类似于复杂随机矩阵的属展开的组合方法进行矩阵计算，但其中出现了不可定向的表面，证明了真实模型之间的共性以及与其复杂类似物的区别，从而激发了这种独特的定义。在实际情况中，我们发现，除了出现在与环形辐条图相对应的复杂情况中的项之外，还有一组与环形辐条图相对应的额外项，其中环面的两个圆方向相反，并且其中出现矩阵转置。
We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, three real models of random matrices, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. We use a combinatorial approach to the matrix calculations similar to the genus expansion for complex random matrices, but in which nonorientable surfaces appear, demonstrating the commonality between the real models and the distinction from their complex analogues, motivating this distinct definition. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears.