A Course on Large Deviations with an Introduction to Gibbs Measures Firas Rassoul-Agha Timo Seppäläinen Graduate Studies in Mathematics Volume 162 American Mathematical Society A Course on Large Deviations with an Introduction to Gibbs Measures https://doi.org/10.1090//gsm/162 A Course on Large Deviations with an Introduction to Gibbs Measures Firas Rassoul-Agha Timo Seppäläinen Graduate Studies in Mathematics Volume 162 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 60-01, 60F10, 60J10, 60K35, 60K37, 82B05, 82B20. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-162 Library of Congress Cataloging-in-Publication Data Rassoul-Agha, Firas. A course on large deviations with an introduction to Gibbs measures / Firas Rassoul-Agha, Timo Sepp¨al¨ainen. pages cm. — (Graduate studies in mathematics ; volume 162) Includes bibliographical references and indexes. ISBN 978-0-8218-7578-0 (alk. paper) 1. Large deviations. 2. Probabilities. 3. Measure theory. I. Sepp¨al¨ainen, Timo O. II. Title. QA273.67.R375 2015 519.2—dc23 2014035275 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. 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Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 To Alla, Maxim, and Kirill To Celeste, David, Ansa, and Timo Contents Preface xi Part I. Large deviations: General theory and i.i.d. processes Chapter 1. Introductory discussion 3 §1.1. Information-theoretic entropy 5 §1.2. Thermodynamic entropy 8 §1.3. Large deviations as useful estimates 12 Chapter 2. The large deviation principle 17 §2.1. Precise asymptotics on an exponential scale 17 §2.2. Lower semicontinuous and tight rate functions 20 §2.3. Weak large deviation principle 23 §2.4. Aspects of Cram´er’s theorem 26 §2.5. Limits, deviations, and fluctuations 33 Chapter 3. Large deviations and asymptotics of integrals 35 §3.1. Contraction principle 35 §3.2. Varadhan’s theorem 37 §3.3. Bryc’s theorem 41 §3.4. Curie-Weiss model of ferromagnetism 43 Chapter 4. Convex analysis in large deviation theory 49 §4.1. Some elementary convex analysis 49 §4.2. Rate function as a convex conjugate 58 §4.3. Multidimensional Cram´er theorem 61 vii viii Contents Chapter 5. Relative entropy and large deviations for empirical measures 67 §5.1. Relative entropy 67 §5.2. Sanov’s theorem 73 §5.3. Maximum entropy principle 78 Chapter 6. Process level large deviations for i.i.d. fields 83 §6.1. Setting 83 §6.2. Specific relative entropy 85 §6.3. Pressure and the large deviation principle 91 Part II. Statistical mechanics Chapter 7. Formalism for classical lattice systems 99 §7.1. Finite-volume model 99 §7.2. Potentials and Hamiltonians 101 §7.3. Specifications 103 §7.4. Phase transition 108 §7.5. Extreme Gibbs measures 110 §7.6. Uniqueness for small potentials 112 Chapter 8. Large deviations and equilibrium statistical mechanics 121 §8.1. Thermodynamic limit of the pressure 121 §8.2. Entropy and large deviations under Gibbs measures 124 §8.3. Dobrushin-Lanford-Ruelle (DLR) variational principle 127 Chapter 9. Phase transition in the Ising model 133 §9.1. One-dimensional Ising model 136 §9.2. Phase transition at low temperature 138 §9.3. Case of no external field 141 §9.4. Case of nonzero external field 146 Chapter 10. Percolation approach to phase transition 149 §10.1. Bernoulli bond percolation and random cluster measures 149 §10.2. Ising phase transition revisited 153 Contents ix Part III. Additional large deviation topics Chapter 11. Further asymptotics for i.i.d. random variables 161 §11.1. Refinement of Cram´er’s theorem 161 §11.2. Moderate deviations 164 Chapter 12. Large deviations through the limiting generating function 167 §12.1. Essential smoothness and exposed points 167 §12.2. G¨artner-Ellis theorem 175 §12.3. Large deviations for the current of particles 179 Chapter 13. Large deviations for Markov chains 187 §13.1. Relative entropy for kernels 187 §13.2. Countable Markov chains 191 §13.3. Finite Markov chains 203 Chapter 14. Convexity criterion for large deviations 213 Chapter 15. Nonstationary independent variables 221 §15.1. Generalization of relative entropy and Sanov’s theorem 221 §15.2. Proof of the large deviation principle 223 Chapter 16. Random walk in a dynamical random environment 233 §16.1. Quenched large deviation principles 234 §16.2. Proofs via the Baxter-Jain theorem 239 Appendixes Appendix A. Analysis 259 §A.1. Metric spaces and topology 259 §A.2. Measure and integral 262 §A.3. Product spaces 267 §A.4. Separation theorem 268 §A.5. Minimax theorem 269 x Contents Appendix B. Probability 273 §B.1. Independence 274 §B.2. Existence of stochastic processes 275 §B.3. Conditional expectation 276 §B.4. Weak topology of probability measures 278 §B.5. First limit theorems 282 §B.6. Ergodic theory 282 §B.7. Stochastic ordering 288 Appendix C. Inequalities from statistical mechanics 293 §C.1. Griffiths’s inequality 293 §C.2. Griffiths-Hurst-Sherman inequality 294 Appendix D. Nonnegative matrices 297 Bibliography 299 Notation index 305 Author index 311 General index 313 Preface This book arose from courses on large deviations and related topics given by the authors in the Departments of Mathematics at the Ohio State University (1993), at the University of Wisconsin-Madison (2006 and 2013), and at the University of Utah (2008 and 2013). Our goal has been to create an attractive collection of material for a semester’s course which would also serve the broader needs of students from different fields. This goal has had two implications for the book. (1) We have not aimed at anything like an encyclopedic coverage of different techniques for proving large deviation principles (LDPs). Part I of the book focuses on one classic line of reasoning: (i) upper bound by an exponential Markov-Chebyshev inequality, (ii) lower bound by a change of measure, and (iii) an argument to match the rates from (i) and (ii). Beyond this technique Part I covers Bryc’s theorem and proves Cram´er’s theorem with the subadditive method. Part III of the book covers the G¨artner-Ellis theorem and an approach based on the convexity of a local rate function due to Baxter and Jain. (2) We have not felt obligated to stay within the boundaries of large deviation theory but instead follow the trail of some interesting material. Large deviation theory is a natural gateway to statistical mechanics. A dis- cussion of statistical mechanics would be incomplete without some study of phase transitions. We prove the phase transition of the Ising model in two different ways: (i) first with classical techniques: the Peierls argument, Dobrushin’s uniqueness condition, and correlation inequalities and (ii) the second time with random cluster measures. This means leaving large de- viation theory completely behind. Along the way we have the opportunity to learn coupling methods which are central to modern probability theory xi xii Preface but do not get serious application in the typical first graduate course in probability. We give now a brief overview of the contents of the book. Part I covers core general large deviation theory, the relevant convex analysis, and the large deviations of independent and identically distributed (i.i.d.) processes on three levels: Cram´er’s theorem, Sanov’s theorem, and the process level LDP for i.i.d. variables indexed by a multidimensional square lattice. Part II introduces Gibbs measures and proves the Dobrushin-Lanford- Ruelle variational principle that characterizes translation-invariant Gibbs measures. After this we study the phase transition of the Ising model. Part II ends with a chapter on the Fortuin-Kasteleyn random cluster model and the percolation approach to Ising phase transition. Part III develops the large deviation themes of Part I in several direc- tions. Large deviations of i.i.d. variables are complemented with moderate deviations and with more precise large deviation asymptotics. The G¨artner- Ellis theorem is developed carefully, together with the necessary additional convex analysis beyond the basics covered in Part I. From large deviations of i.i.d. processes we move on to Markov chains, to nonstationary indepen- dent random variables, and finally to random walk in a dynamical random environment. The last two topics give us an opportunity to apply another approach to proving large deviation principles, namely the Baxter-Jain the- orem.