We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behavior of unitarily invariant and orthogonally invariant random matrices respectively. Most notably, because the quaternionic trace does not have the cyclic property of a trace over a commutative field, the asymmetries which appear in the multi-matrix context result in an asymmetric contribution from the terms which appear symmetrically in the complex and real cases.

中文翻译：

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四元数二阶自由度与大辛不变随机矩阵的涨落

我们提出了四元数情况下二阶自由度的定义。我们证明了二阶概率空间上的这个定义被独立的辛不变四元数矩阵渐近地满足。这个定义不同于复数和实数二阶概率空间的自然定义，它们分别由酉不变和正交不变随机矩阵的渐近行为驱动。最值得注意的是，因为四元数迹不具有交换域上迹的循环特性，出现在多矩阵上下文中的不对称性导致在复杂和真实情况下对称出现的项的不对称贡献。