Asymptotic Expansion of a Partition Function Related to the Sinh-model
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 introducti
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Gaëtan Borot Alice Guionnet Karol K. Kozlowski
Asymptotic Expansion of a Partition Function Related to the Sinh-model
Mathematical Physics Studies Series editors Giuseppe Dito, Dijon, France Edward Frenkel, Berkeley, CA, USA Sergei Gukov, Pasadena, CA, USA Yasuyuki Kawahigashi, Tokyo, Japan Maxim Kontsevich, Bures-sur-Yvette, France Nicolaas P. Landsman, Nijmegen, The Netherlands
More information about this series at http://www.springer.com/series/6316
Gaëtan Borot Alice Guionnet Karol K. Kozlowski •
Asymptotic Expansion of a Partition Function Related to the Sinh-model
123
Karol K. Kozlowski ENS de Lyon Laboratoire de Physique-UMR 5672 du CNRS Lyon France
Gaëtan Borot Max Planck Institut für Mathematik Bonn Germany Alice Guionnet Department of Mathematics MIT Cambridge, MA USA
ISSN 0921-3767 Mathematical Physics Studies ISBN 978-3-319-33378-6 DOI 10.1007/978-3-319-33379-3
ISSN 2352-3905
(electronic)
ISBN 978-3-319-33379-3
(eBook)
Library of Congress Control Number: 2016939912 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
An opening discussion
The present work develops techniques enabling one to carry out the largeN asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh-model: zN ½ W ¼
Z Y N RN
a\b
fsinh½px1 ðya yb Þsinh½px2 ðya yb Þgb
N Y
eW ðya Þ dN y:
a¼1
When b ¼ 1 and for specific choices of the constants x1 ; x2 [ 0 and of the confining potential W, zN represents norms or arises as a fundamental building block of certain classes of correlation functions in quantum integrable models that are solvable by the quantum separation of
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