Random matrices and non-commutative probability
著者
書誌事項
Random matrices and non-commutative probability
CRC Press, 2022
大学図書館所蔵 全10件
注記
Includes bibliographical references (p. 251-258) and index
内容説明・目次
内容説明
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 Moebius 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 Moebius 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.
目次
Classical independence, moments and cumulants. 2. Non-commutative probability. 3. Free independence. 4. Convergence. 5. Transforms. 6. C* -probability space. 7. Random matrices. 8. Convergence of some important matrices. 9. Joint convergence I: single pattern. 10. Joint convergence II: multiple patterns. 11. Asymptotic freeness of random matrices. 12. Brown measure. 13. Tying three loose ends.
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