论文标题
通用包裹的交通空间中的刚性结构
Rigid structures in the universal enveloping traffic space
论文作者
论文摘要
对于任何奇特的非交通概率空间$（\ Mathcal {a}，φ）$，cébron，dahlqvist和Male表明，人们总是可以构造一个包裹着的交通空间$（\ Mathcal {g}（\ Mathcal {a}），τi_φ），τive）$。该构建提供了一个通用的对象，使人们可以在通用情况下吸引流量概率框架，从而优先考虑其结构。在本文中，我们证明$（\ Mathcal {g}（\ Mathcal {a}），τ_φ）$允许一个规范的免费产品分解$ \ MATHCAL {a} * \ MATHCAL {a} a} a}^^^\ intercal *θ（\ mathcal {\ mathcal {g} { In particular, $\mathcal{A}^\intercal$ is an anti-isomorphic copy of $\mathcal{A}$, and $Θ(\mathcal{G}(\mathcal{A}))$ is, up to degeneracy, a commutative algebra generated by Gaussian random variables with a covariance structure diagonalized by the graph 运营。如果$（\ Mathcal {a}，φ）$本身是一个免费产品，那么我们描述了该附加结构如何将其提升为$（\ Mathcal {g}（\ Mathcal {a}），τ_φ）$。在这里，我们发现自由独立与通常方向相对的古典独立之间的联系。直到退化，我们进一步表明$（\ Mathcal {g}（\ Mathcal {a}），τ_φ）$由树状图形操作跨越。最后，我们将结果应用于大型（可能取决于）随机矩阵的研究。我们的分析依赖于仙人掌图的组合和所得的仙人掌肿瘤对应关系。
For any tracial non-commutative probability space $(\mathcal{A}, φ)$, Cébron, Dahlqvist, and Male showed that one can always construct an enveloping traffic space $(\mathcal{G}(\mathcal{A}), τ_φ)$ that extends the trace. This construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. In this article, we prove that $(\mathcal{G}(\mathcal{A}), τ_φ)$ admits a canonical free product decomposition $\mathcal{A} * \mathcal{A}^\intercal * Θ(\mathcal{G}(\mathcal{A}))$. In particular, $\mathcal{A}^\intercal$ is an anti-isomorphic copy of $\mathcal{A}$, and $Θ(\mathcal{G}(\mathcal{A}))$ is, up to degeneracy, a commutative algebra generated by Gaussian random variables with a covariance structure diagonalized by the graph operations. If $(\mathcal{A}, φ)$ itself is a free product, then we describe how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), τ_φ)$. Here, we find a connection between free independence and classical independence opposite the usual direction. Up to degeneracy, we further show that $(\mathcal{G}(\mathcal{A}), τ_φ)$ is spanned by tree-like graph operations. Finally, we apply our results to the study of large (possibly dependent) random matrices. Our analysis relies on the combinatorics of cactus graphs and the resulting cactus-cumulant correspondence.
