Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence (Memoirs of the American Mathematical Society) - Softcover

Male, Camille

Softcover

ISBN 10: 1470442981 ISBN 13: 9781470442989

Publisher: Amer Mathematical Society, 2021

Synopsis

Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability . The author first establishes this construction for random matrices and then defines the traffic distribution of random matrices, which is richer than 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 calls traffic independence the asymptotic rule that plays the role of independence with respect to traffic distributions. Wigner matrices, Haar unitary matrices and uniform permutation matrices converge in traffic distributions, a fact which yields new results on the limiting $^*$-distributions of several matrices the author can construct from them. Then the author defines the abstract traffic spaces as non commutative probability spaces with more structure. She proves that at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free and Boolean independence. A central limiting theorem is stated in this context, interpolating between the tensor, free and Boolean central limit theorems.

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2021
• Language: English
• ISBN 10: 1470442981
• ISBN 13: 9781470442989
• Binding: Paperback
• Number of pages: 88