Rigid structures in the universal enveloping traffic space

2020/11/11 by Benson Au, Au, Benson, Camille Male +1
Mathematics · #15B52 #46L53 #46L54 #60B20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics #Combinatorics (math.CO) #Commutative property #Computer science #Connection (principal bundle) #Degeneracy (biology) #Discrete mathematics #FOS: Mathematics #Free product #Geometry #Group (periodic table) #Mathematics #Operator Algebras (math.OA) #Physics #Probability (math.PR) #Product (mathematics) #Quantum mechanics #Random Matrices and Applications #Space (punctuation) #math.CO #math.OA #math.PR #msc:15B52 #msc:46L53 #msc:46L54 #msc:60B20

paper · pdf · doi:10.48550/arxiv.2011.05472

54 pages, 16 figures

arxiv created 2020/11/11 · openalex publication_date 2020/11/11 · arxiv updated 2020/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any tracial non-commutative probability space (A, φ), Cébron, Dahlqvist, and Male showed that one can always construct an enveloping traffic space (G(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 (G(A), τ_φ) admits a canonical free product decomposition A * A^\intercal * Θ(G(A)). In particular, A^\intercal is an anti-isomorphic copy of A, and Θ(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, φ) itself is a free product, then we describe how this additional structure lifts into (G(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 (G(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.

Citations

• Zur allgemeinen Kurventheorie
• Spectral measure of large random Hankel, Markov and Toeplitz matrices
• Limit laws for Random matrices and free products

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