A Course on Large Deviations with an Introduction to Gibbs Measures.

More documents

Recommendations

Info

$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

viii Contents §5.3. Multidimensional Cramér theorem revisited 51 Chapter 6. Relative entropy and large deviations for empirical measures 57 §6.1. Relative entropy 58 §6.2. Sanov’s theorem 62 §6.3. Maximum entropy principle 67 Chapter 7. Large deviations for i.i.d. fields at the process level 71 §7.1. Setting 71 §7.2. Specific relative entropy 73 §7.3. Pressure and the large deviation principle 78 Part II. Statistical Mechanics Chapter 8. Formalism for classical lattice systems 87 §8.1. Finite volume model 87 §8.2. Potentials and Hamiltonians 88 §8.3. Specifications 90 §8.4. Gibbs specifications and phase transition 93 §8.5. Observables 96 Chapter 9. Large deviations and equilibrium statistical mechanics 99 §9.1. Replacing Hamiltonians with averages 99 §9.2. Thermodynamic limit of the pressure 101 §9.3. Specific relative entropy 102 §9.4. Large deviations under Gibbs kernels 104 §9.5. Dobrushin-Lanford-Ruelle (DLR) variational principle 105 Chapter 10. Phase transition in the Ising model 111 §10.1. One-dimensional Ising model 113 §10.2. Phase transition at low temperature 115 §10.3. Uniqueness of phase at high temperature 119 §10.4. Case of no external field 121 §10.5. Case of nonzero external field 124 Chapter 11. Percolation approach to phase transition 129 Part III. Further large deviations topics Chapter 12. Further asymptotics for i.i.d. random variables 133
Contents ix §12.1. Refinement of Cramér’s theorem 133 §12.2. Moderate deviations 136 Chapter 13. Large deviations for Markov chains 141 §13.1. Restricting entropies on product spaces 141 §13.2. Large deviations 144 Chapter 14. Convexity criterion for large deviations 145 Chapter 15. Nonstationary independent variables 153 §15.1. Generalization of relative entropy and Sanov’s theorem 153 §15.2. Proof of the large deviation principle 155 Appendixes Appendix A. Topics from probability 167 §A.1. Weak convergence of probability measures 167 §A.2. Ergodic theorem 170 §A.3. Stochastic ordering 175 Appendix B. Topics from analysis 177 §B.1. Measure-theoretic lemma 177 §B.2. Minimax theorem 178 Appendix C. Inequalities 183 §C.1. Holley’s inequality 183 §C.2. Griffiths’ inequality 185 §C.3. Griffiths-Hurst-Sherman inequality 186 Bibliography 189 Notation index 193 Theorems, principles, and models index 197 Author index 199 General index 201
Page 1 and 2: A Course on Large
Page 3: To Alla, Maxim, and Kirill To Celes
Page 9 and 10: Preface (and to the reviewers) This
Page 11: Preface (and to the reviewers) xiii
Page 15 and 16: Introduction Chapter 1 Imagine the
Page 17 and 18: 1.1. Information-theoretic entropy
Page 19 and 20: 1.2. Thermodynamic entropy 7 to 0.
Page 21: 1.2. Thermodynamic entropy 9 The id
Page 24 and 25: 12 2. Preliminary examples and gene
Page 32 and 33: 20 3. More generalities and Cramér
Page 42 and 43: 30 4. Yet some more generalities Th
Page 44 and 45: 32 4. Yet some more generalities Va
Page 46 and 47: 34 4. Yet some more generalities Va
Page 48 and 49: 36 4. Yet some more generalities fu
Page 50 and 51: 38 4. Yet some more generalities Ha
Page 52 and 53: 40 4. Yet some more generalities Ne
Page 54 and 55: 42 5. Convex analysis in large devi
Page 56 and 57:
44 5. Convex analysis in large devi
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
58 6. Relative entropy and large de
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
72 7. Large deviations for i.i.d. f
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 99 and 100:
Formalism for classical lattice sys
Page 101 and 102:
8.2. Potentials and Hamiltonians 89
Page 103 and 104:
8.3. Specifications 91 Example 8.5.
Page 105 and 106:
8.4. Gibbs specifications and phase
Page 107 and 108:
8.4. Gibbs specifications and phase
Page 109 and 110:
8.5. Observables 97 In either case,
Page 111 and 112:
Large deviations and equilibrium st
Page 113 and 114:
9.2. Thermodynamic limit of the pre
Page 115 and 116:
9.3. Specific relative entropy 103
Page 117 and 118:
9.5. Dobrushin-Lanford-Ruelle (DLR)
Page 119 and 120:
Page 121:
Page 124 and 125:
112 10. Phase transition in the Isi
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 141:
Percolation approach to phase trans
Page 145 and 146:
Further asymptotics for i.i.d. rand
Page 147 and 148:
12.1. Refinement of Cramér’s the
Page 149 and 150:
12.2. Moderate deviations 137 Theor
Page 151:
12.2. Moderate deviations 139 where
Page 154 and 155:
142 13. Large deviations for Markov
Page 156 and 157:
144 13. Large deviations for Markov
Page 158 and 159:
146 14. Convexity criterion for lar
Page 160 and 161:
Page 162 and 163:
Page 165 and 166:
Nonstationary independent variables
Page 167 and 168:
15.2. Proof of the large deviation
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175:
Page 179 and 180:
Topics from probability Appendix A
Page 181 and 182:
A.1. Weak convergence of probabilit
Page 183 and 184:
A.2. Ergodic theorem 171 To illumin
Page 185 and 186:
A.2. Ergodic theorem 173 measurable
Page 187 and 188:
A.3. Stochastic ordering 175 de Fin
Page 189 and 190:
Topics from analysis Ultimately, th
Page 191 and 192:
B.2. Minimax theorem 179 ∗ Exerci
Page 193:
B.2. Minimax theorem 181 Now, if
Page 196 and 197:
184 C. Inequalities Next, let C = m
Page 198 and 199:
186 C. Inequalities The inequality
Page 201 and 202:
Bibliography 1. Michael Aizenman, I
Page 203 and 204:
Bibliography 191 24. Y. Higuchi, On
Page 205 and 206:
Notation index empirical mean (X1 +
Page 207:
Notation index 195 B space of absol
Page 211:
Aizenman, 113 Bartle, 6 Baxter and
Page 214 and 215:
202 General index average, 99 field
Page 216 and 217:
204 General index proper function,
show all

A Course on Large Deviations with an Introduction to Gibbs Measures.

Create successful ePaper yourself

Delete template?

Save as template?