Softcover ISBN:  9781470450274 
Product Code:  CBMS/130 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
eBook ISBN:  9781470451943 
Product Code:  CBMS/130.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
Softcover ISBN:  9781470450274 
eBook: ISBN:  9781470451943 
Product Code:  CBMS/130.B 
List Price:  $116.00 $87.00 
MAA Member Price:  $104.40 $78.30 
AMS Member Price:  $92.80 $69.60 
Softcover ISBN:  9781470450274 
Product Code:  CBMS/130 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
eBook ISBN:  9781470451943 
Product Code:  CBMS/130.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
Softcover ISBN:  9781470450274 
eBook ISBN:  9781470451943 
Product Code:  CBMS/130.B 
List Price:  $116.00 $87.00 
MAA Member Price:  $104.40 $78.30 
AMS Member Price:  $92.80 $69.60 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 130; 2019; 143 ppMSC: Primary 60; 46
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.
ReadershipGraduate students and researchers interested in applications to the asymptotic analysis of DysonSchwinger equations.

Table of Contents

Chapters

Introduction

The example of the GUE

Wigner random matrices

Betaensembles

Discrete betaensembles

Continuous betamodels: The several cut case

Several matrixensembles

Universality for betamodels


Additional Material

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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.
Graduate students and researchers interested in applications to the asymptotic analysis of DysonSchwinger equations.

Chapters

Introduction

The example of the GUE

Wigner random matrices

Betaensembles

Discrete betaensembles

Continuous betamodels: The several cut case

Several matrixensembles

Universality for betamodels