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