Rigid structures in traffic probability: with a view toward random matrices

Traffic probability is an operadic non-commutative probability theory recently introduced by Male that generalizes the standard non-commutative probabilistic framework. This additional operad structure admits a corresponding notion of independence, the so-called \emph{traffic independence}. At the same time, traffic probability captures certain aspects of both classical and free probability. An as yet incomplete understanding of this relationship yields insightful feedback between the different theories. In this dissertation, we study this problem through two complementary angles: first, in the context of the universal enveloping traffic space; and second, in the context of large random matrices. For a tracial non-commutative probability space $(\mathcal{A}, \varphi)$, C\'{e}bron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace. The CDM 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 Chapter 3, we study the structure of the universal enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ as a general non-commutative probability space $(\mathcal{B}, \psi)$, particularly in relation to non-commutative notions of independence. We show that $(\mathcal{B}, \psi)$ admits a canonical free product decomposition $\mathcal{B} = \mathcal{A} * \mathcal{A}^\intercal * \Theta(\mathcal{B})$, regardless of the choice of $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{B}, \psi)$. Here, we find a duality between classical independence and free independence. Our proof relies on the existence of a natural homomorphic conditional expectation in $(\mathcal{B}, \psi)$ that takes $\Theta(\mathcal{B})$ to a commutative subalgebra $\Delta(\mathcal{B})$. Up to degeneracy, we further show that $\Delta(\mathcal{B})$ is spanned by tree-like graph operations. In Chapter 4, we utilize the traffic framework to study the asymptotics of large random multi-matrix models. As a starting point, we compute the limiting traffic distribution of the classical ensembles of Wigner, Ginibre, and Wishart-Laguerre. This allows us to apply our free product decomposition from Chapter 3 to prove the asymptotic freeness of a large class of dependent random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. We further prove general Markov-type concentration inequalities for the joint traffic distribution of our matrices. We then extend our analysis to random band matrices and investigate the extent to which the joint traffic distribution of these matrices deviates from the classical case. We also pursue an orthogonal computation, namely, that of a Haar distributed orthogonal random matrix. Altogether, our formulas suggest a convenient cactus-cumulant correspondence, the details of which we commit in the last section. Our results related to the universal enveloping traffic space form part of a joint work in progress with Camille Male.

Advisor: Evans, Steven N