An Introduction to Random MatricesCambridge University Press, 2010 - 492 Seiten The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence. |
Inhalt
Real and complex Wigner matrices | 6 |
logarithmic Sobolev inequalities | 38 |
Hermite polynomials spacings and limit distributions for the Gaus | 90 |
6 1 | 126 |
Some generalities | 186 |
with independent entries | 282 |
Free probability | 322 |
laws | 325 |
Appendices | 414 |
G Appendix on operator algebras | 450 |
References | 465 |
General conventions and notation | 481 |
Häufige Begriffe und Wortgruppen
assume assumption asymptotics Borel bounded Brownian motion C*-algebra compact compactly supported completes the proof consider constant converges Corollary deduce defined Definition denote derive determinantal process differential eigenvalues elements empirical measure ensembles equality equation Exercise exists field find finite first fix fixed Fredholm determinant free probability function f Gaussian hence Hermitian Hilbert space identity implies independent inequality joint distribution kernel large deviation Lebesgue measure Lemma limit linear Lipschitz manifold MatN non-crossing nonnegative norm notation Note obtain operator orthogonal partition point process Polish space polynomials positive integers probability measure probability space proof of Lemma proof of Proposition proof of Theorem properties prove random matrices random variables Recall resp right side satisfies Section self-adjoint sequence spectral Stieltjes transform subset tion topology tracial uniformly unique unitary vanishes vector space von Neumann algebra Weyl quadruple Wigner matrices