Asymptotics of random matrices and related models : the uses of DysonSchwinger equations
 Responsibility
 Alice Guionnet.
 Publication
 [Providence, RI] : American Mathematical Society, [2019]
 Physical description
 vii, 143 pages ; 26 cm.
 Series
 Regional conference series in mathematics ; no. 130.
At the library
Science Library (Li and Ma)
Stacks
Call number  Status 

QA196.5 .G85 2019  In process Request 
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Description
Creators/Contributors
 Author/Creator
 Guionnet, Alice, author.
 Contributor
 Conference Board of the Mathematical Sciences.
 National Science Foundation (U.S.)
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 137141) and index.
 Contents

 Introduction The example of the GUE Wigner random matrices Betaensembles Discrete betaensembles Continuous betamodels: The several cut case Several matrixensembles Universality for betamodels Bibliography Index.
 (source: Nielsen Book Data)
 Summary

Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of DysonSchwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models.
(source: Nielsen Book Data)
Subjects
 Subject
 Random matrices.
 Matrices.
 Green's functions.
 Lagrange equations.
 Green's functions.
 Lagrange equations.
 Matrices.
 Random matrices.
 Probability theory and stochastic processes  Probability theory on algebraic and topological structures  Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
 Functional analysis  Selfadjoint operator algebras (algebras, von Neumann algebras, etc.)  Free probability and free operator algebras.
 Probability theory and stochastic processes  Limit theorems  Central limit and other weak theorems.
 Probability theory and stochastic processes  Limit theorems  Large deviations.
Bibliographic information
 Publication date
 2019
 Series
 CBMS regional conference series in mathematics ; number 130
 Note
 "Support from the National Science Foundation."
 ISBN
 9781470450274 paperback
 1470450275 paperback