Traffic distributions and independence : permutation invariant random matrices and the three notions of independence /

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 ex...

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Bibliographic Details
Author / Creator:Male, Camille, 1984- author.
Imprint:Providence, RI : American Mathematical Society, [2020]
©2020.
Description:v, 88 pages : illustrations ; 26 cm.
Language:English
Series:Memoirs of the American Mathematical Society, 0065-9266 ; number 1300
Memoirs of the American Mathematical Society ; no. 1300.
Subject:Random matrices.
Independence (Mathematics)
Asymptotic distribution (Probability theory)
Limit theorems (Probability theory)
Selfadjoint operators.
Selfadjoint operators.
Limit theorems (Probability theory)
Independence (Mathematics)
Asymptotic distribution (Probability theory)
Random matrices.
Theory of distributions (Functional analysis)
Traffic flow -- Mathematical models.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12617465
Hidden Bibliographic Details
ISBN:9781470442989
1470442981
9781470463991
Notes:"September 2020, volume 267, number 1300 (fourth of 7 numbers)."
Includes bibliographical references.
Summary: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.