Asymptotic Expansion of a Partition Function Related to the Sinh-model

1. Front Matter

   Pages i-xv

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2. Introduction

   • Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

   Pages 1-52

3. Main Results and Strategy of Proof

   • Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

   Pages 53-67

4. Asymptotic Expansion of —The Schwinger–Dyson Equation Approach

   • Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

   Pages 69-98

5. The Riemann–Hilbert Approach to the Inversion of

   • Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

   Pages 99-131

6. The Operators

   • Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

   Pages 133-164

7. Asymptotic Analysis of Integrals

   • Gaëtan Borot, Alice Guionnet, Karol K. Kozlowski

   Pages 165-192

8. Back Matter

   Pages 193-222

   PDF

This book elaborates on the asymptotic behaviour, when N is large, of certain N-dimensional integrals which typically occur in random matrices, or in 1+1 dimensional quantum integrable models solvable by the quantum separation of variables. The introduction presents the underpinning motivations for this problem, a historical overview, and a summary of the strategy, which is applicable in greater generality. The core aims at proving an expansion up to o(1) for the logarithm of the partition function of the sinh-model. This is achieved by a combination of potential theory and large deviation theory so as to grasp the leading asymptotics described by an equilibrium measure, the Riemann-Hilbert approach to truncated Wiener-Hopf in order to analyse the equilibrium measure, the Schwinger-Dyson equations and the boostrap method to finally obtain an expansion of correlation functions and the one of the partition function. This book is addressed to researchers working in random matrices, statistical physics or integrable systems, or interested in recent developments of asymptotic analysis in those fields.

• Schwinger-Dyson equation
• Riemann-Hilbert problem
• Gaussian potential
• concentration of measure
• loop equations
• KPZ models
• Toda lattice
• six-vertex model
• XXZ chains
• algebraic Bethe Ansatz
• quantum Toda chain
• Selberg integral
• separation of variables
• quantum separation of variables
• random matrix theory

“The main task of the book is to develop an effective method to obtain asymptotic expansions for certain rescaled multiple integrals. … The book contains five appendices which complement the main results obtained. The book is addressed to researchers working in random matrices, statistical physics or integrable systems, or interested in recent developments of asymptotic analysis in those fields.” (Horacio Grinberg, Mathematical Reviews, August, 2017)

• Max Planck Institut für Mathematik, Bonn, Germany

  Gaëtan Borot

• Department of Mathematics, MIT, Cambridge, USA

  Alice Guionnet

• ENS de Lyon, Laboratoire de Physique-UMR 5672 du CNRS, Lyon, France

  Karol K. Kozlowski

Gaëtan Borot graduated at ENS Paris in theoretical physics, did his PhD at CEA Saclay, and is now a W2 Group Leader at the Max Planck Institute for Mathematics in Bonn. He was also a visiting scholar at MIT, collaborating with Alice Guionnet on the asymptotic analysis of random matrix models. He is working on the mathematical aspects of geometry and physics, ranging from statistical physics, random matrices, integrable systems, enumerative geometry, topological quantum field theories, etc.


Alice Guionnet is Director of research CNRS at École Normale Supérieure (ENS) Lyon, from MIT where she served as a professor in 2012-2015. She received the MS from ENS Paris in 1993 and the PhD, under the guidance of G. Ben Arous at Université Paris Sud in 1995.
A. Guionnet is a world leading probabilist, working on a program related to operator algebra theory and mathematical physics. 
She has made important contributions in random matrix theory,including large deviations, topological expansions, but also more classical study of their spectrum and eigenvectors. From 2006-2011 she served as Editor-in-Chief of Annales de L’Institut Henri Poincaré (currently on its editorial board), and also serves on the editorial board of Annals of Probability.
She has given two Plenary talks and a number of Invited Talks at international meetings, including ICM. Her distinctions include the Miller Institute Fellowship, (2006), the Loève Prize (2009), the Silver Medal of CNRS (2010) and Simon Investigator (2012).


Karol Kajetan Kozlowski is a CNRS Chargé de recherche at the École Normale Supérieure (ENS) Lyon. 
He graduated from ENS-Lyon in 2005 and did his PhD at the Laboratoire Physique of ENS-Lyon. He was then a post-doctoral fellow at the Deutsches Elektronen-Synchrotron. His main research interest concern quantum integrable models and various aspects of asymptotic analysis.

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