Traffic Distributions And Independence Permutation Invariant Random Matrices And The Three Notions Of Independence
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Traffic Distributions and Independence: Permutation Invariant Random Matrices and the Three Notions of Independence
Author: Camille Male
language: en
Publisher: American Mathematical Society
Release Date: 2021-02-10
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.
Random Matrices and Non-Commutative Probability
This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful. Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability. Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants. Free cumulants are introduced through the Möbius function. Free product probability spaces are constructed using free cumulants. Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed. Convergence of the empirical spectral distribution is discussed for symmetric matrices. Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices. Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. Exercises, at advanced undergraduate and graduate level, are provided in each chapter.
The Yang-Mills Heat Equation with Finite Action in Three Dimensions
Author: Leonard Gross
language: en
Publisher: American Mathematical Society
Release Date: 2022-02-02
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