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Stochastic Processes and Random Matrices
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Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School: Volume 104, July 2015

Grégory Schehr, Alexander Altland, Yan V. Fyodorov, Neil O'Connell, and Leticia F. Cugliandolo

Abstract

The field of stochastic processes and random matrix theory (RMT) has been a rapidly evolving subject during the past fifteen years where the continuous development and discovery of new tools, connections, and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances ... More

Keywords: random matrix theory, stochastic processes, Kardar–Parisi–Zhang, free probability, integrable probability, loop equations, multiple input-multiple output (MIMO) systems, number theory, random Schrödinger operators

Bibliographic Information

Print publication date: 2017 Print ISBN-13: 9780198797319
Published to Oxford Scholarship Online: January 2018 DOI:10.1093/oso/9780198797319.001.0001

Authors

Affiliations are at time of print publication.

Grégory Schehr, editor
Senior Researcher, CNRS - Université Paris-Sud, France

Alexander Altland, editor
Professor of Theoretical Condensed Matter Physics, Institute for Theoretical Physics, University of Köln, Germany

Yan V. Fyodorov, editor
Professor, Department of Mathematics, King's College London, UK

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Contents

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1 History—an overview

Oriol Bohigas1 and Hans A. Weidenmüller2 1LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France; deceased 2Max-Planck-Institut für Kemphysik, Heidelberg, P.O. Box 103980, 69029 Heidelberg, Germany

2 Integrable probability: stochastic vertex models and symmetric functions

Alexei Borodin1 and Leonid Petrov2 1Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA; and Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia 2Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA; and Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia

3 Free probability

Alice Guionnet, ENS Lyon, Université de Lyon, UMPA, UMR 5669 CNRS, 46 allée d’Italie, 69364 Lyon Cedex 07, France

4 The Kardar–Parisi–Zhang equation: a statistical physics perspective

Herbert Spohn, zentrum Mathematik and Physik Department, Technische Universität München, Boltzmannstraße 3, 85747 Garching, Germany

5 Random matrix theory and quantum chromodynamics

Gernot Akemann, faculty of Physics, Bielefeld University, Postfach 100131, D-33501 Bielefeld, Germany

6 Random matrix theory and (big) data analysis

Jean-Philippe Bouchaud, capital Fund Management, 23 rue de l’Université, 75 007 Paris, France

7 Random matrices and loop equations

Bertrand Eynard, IPHT/CEA/Saclay, France, and CRM, Montréal, Canada

8 Random matrices and number theory: some recent themes

Jon P. Keating, school of Mathematics, University of Bristol, Bristol BS8 1TW, UK

9 Modern telecommunications: a playground for physicists?

Aris L. Moustakas, department of Physics, National and Kapodistrian University of Athens, Greece

10 Random matrix approaches to open quantum systems

Henning Schomerus, department of Physics, Lancaster University, Lancaster, LA1 4YB, UK

11 Impurity models and products of random matrices

Alain Comtet1 and Yves Tourigny2 1LPTMS, Université Paris 11, 91400, Orsay, France, Sorbonne Universités, UPMC, Université Paris 06, 70005, Paris, France, 2School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

12 Gaussian multiplicative chaos and Liouville quantum gravity

Rémi Rhodes1 and Vincent Vargas2 1Université Paris-Est Marne la Vallée, LAMA, Champs sur Marne, France, and 2ENS Ulm, DMA, 45 rue d’Ulm, 75005 Paris, France

13 Quantum spin chains and classical integrable systems

Anton Zabrodin, institute of Biochemical Physics RAS, 4 Kosygina st., Moscow 119334, Russia; and ITEP, 25 B.Cheremushkinskaya, Moscow 117218, Russia; and Laboratory of Mathematical Physics, National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia