Rigid structures in the universal enveloping traffic space

Abstract

For any tracial non-commutative probability space $(\mathcal{A}, \varphi)$, C\'{e}bron, Dahlqvist, and Male showed that one can always construct an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ 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 $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ admits a canonical free product decomposition $\mathcal{A} * \mathcal{A}^\intercal * \Theta(\mathcal{G}(\mathcal{A}))$. In particular, $\mathcal{A}^\intercal$ is an anti-isomorphic copy of $\mathcal{A}$, and $\Theta(\mathcal{G}(\mathcal{A}))$ is, up to degeneracy, a commutative algebra generated by Gaussian random variables with a covariance structure diagonalized by the graph operations. If $(\mathcal{A}, \varphi)$ itself is a free product, then we describe how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a connection between free independence and classical independence opposite the usual direction. Up to degeneracy, we further show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ 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.