Skew-orthogonal Polynomials and Random Matrix Theory
Title | Skew-orthogonal Polynomials and Random Matrix Theory PDF eBook |
Author | Saugata Ghosh |
Publisher | |
Pages | 127 |
Release | 2009 |
Genre | Electronic books |
ISBN | 9781470417710 |
Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, de.
Skew-orthogonal Polynomials and Random Matrix Theory
Title | Skew-orthogonal Polynomials and Random Matrix Theory PDF eBook |
Author | Saugata Ghosh |
Publisher | American Mathematical Soc. |
Pages | 138 |
Release | |
Genre | Mathematics |
ISBN | 0821869884 |
"Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the use of the GCD promises to be efficient. Titles in this series are co-published with the Centre de Recherches Mathématiques."--Publisher's website.
Random Matrices
Title | Random Matrices PDF eBook |
Author | Madan Lal Mehta |
Publisher | Elsevier |
Pages | 707 |
Release | 2004-10-06 |
Genre | Mathematics |
ISBN | 008047411X |
Random Matrices gives a coherent and detailed description of analytical methods devised to study random matrices. These methods are critical to the understanding of various fields in in mathematics and mathematical physics, such as nuclear excitations, ultrasonic resonances of structural materials, chaotic systems, the zeros of the Riemann and other zeta functions. More generally they apply to the characteristic energies of any sufficiently complicated system and which have found, since the publication of the second edition, many new applications in active research areas such as quantum gravity, traffic and communications networks or stock movement in the financial markets. This revised and enlarged third edition reflects the latest developements in the field and convey a greater experience with results previously formulated. For example, the theory of skew-orthogoanl and bi-orthogonal polynomials, parallel to that of the widely known and used orthogonal polynomials, is explained here for the first time. - Presentation of many new results in one place for the first time - First time coverage of skew-orthogonal and bi-orthogonal polynomials and their use in the evaluation of some multiple integrals - Fredholm determinants and Painlevé equations - The three Gaussian ensembles (unitary, orthogonal, and symplectic); their n-point correlations, spacing probabilities - Fredholm determinants and inverse scattering theory - Probability densities of random determinants
Random Matrix Theory
Title | Random Matrix Theory PDF eBook |
Author | Percy Deift |
Publisher | American Mathematical Soc. |
Pages | 236 |
Release | 2009-01-01 |
Genre | Mathematics |
ISBN | 0821883577 |
"This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles-orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived." --Book Jacket.
Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights
Title | Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights PDF eBook |
Author | Eli Levin |
Publisher | Springer |
Pages | 168 |
Release | 2018-02-13 |
Genre | Mathematics |
ISBN | 3319729470 |
This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices.
Random Matrix Theory, Interacting Particle Systems and Integrable Systems
Title | Random Matrix Theory, Interacting Particle Systems and Integrable Systems PDF eBook |
Author | Percy Deift |
Publisher | Cambridge University Press |
Pages | 539 |
Release | 2014-12-15 |
Genre | Language Arts & Disciplines |
ISBN | 1107079926 |
This volume includes review articles and research contributions on long-standing questions on universalities of Wigner matrices and beta-ensembles.
The Random Matrix Theory of the Classical Compact Groups
Title | The Random Matrix Theory of the Classical Compact Groups PDF eBook |
Author | Elizabeth S. Meckes |
Publisher | Cambridge University Press |
Pages | 225 |
Release | 2019-08-01 |
Genre | Mathematics |
ISBN | 1108317995 |
This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields.