Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence

Voiculescu's idea of asymptotic free independence is extended for independent random matrices invariant in law by conjugation by permutation matrices, leading to an extension of free probability called traffic probability. This concept unifies tensor, free, and Boolean independence and yields new results on the limiting $^*$-distributions of matrices.

Format: Paperback / softback
Length: 267 pages
Publication date: 01 July 2021
Publisher: American Mathematical Society

Voiculescu's notion of asymptotic free independence, which encompasses a wide range of random matrices, including independent unitary invariant matrices, is renowned. This concept is further extended to independent random matrices that are invariant under conjugation by permutation matrices. This development naturally leads to an extension of free probability, formalized under the notions of traffic probability.

In the author's initial exploration, they establish this construction for random matrices and subsequently define the traffic distribution of random matrices, which surpasses the $^*$-distribution of free probability. The knowledge of the individual traffic distributions of independent permutation invariant families of matrices is sufficient to compute the limiting distribution of the join family.

Under a factorization assumption, the author refers to traffic independence as the asymptotic rule that serves as an independence criterion with respect to traffic distributions. Wigner matrices, Haar unitary matrices, and uniform permutation matrices converge in traffic distributions, yielding novel results on the limiting $^*$-distributions of various matrices that the author can construct from them.

Next, the author defines abstract traffic spaces as non-commutative probability spaces with additional structure. She establishes that, at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free, and Boolean independence. In this context, a central limiting theorem is stated, interpolating between the tensor, free, and Boolean central limit theorems.

Weight: 190g
ISBN-13: 9781470442989