A course on large deviations with an introduction to Gibbs measures:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2015]
|
| Series: | Graduate studies in mathematics
volume 162 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 299-303 |
| Physical Description: | xiv, 318 Seiten Diagramme |
| ISBN: | 9780821875780 |
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Record in the Search Index
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| adam_text | Titel: A course on large deviations with an introduction to Gibbs measures
Autor: Rassoul-Agha, Firas
Jahr: 2015
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 Cramer’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 Cramer theorem 61 vii
Contents viii 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 Cramer’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. Gartner-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.l. 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.l. 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.l. 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
Contents xi Preface 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 Chapter 2. The large deviation principle 12 17 §2.1 . Precise asymptotics on an exponentialscale 17 §2.2 . Lower semicontinuous and tight rate functions 20 §2.3 . Weak large deviation principle 23 §2.4 . Aspects of Cramer’s theorem 26 §2.5 . Limits, deviations, and fluctuations 33 Chapter 3. Large deviations and asymptotics ofintegrals 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 Convex analysis in large deviationtheory Chapter 4. 49 §4.1 . Some elementary convex analysis 49 §4.2 . Rate function as a convex conjugate 58 §4.3 . Multidimensional Cramer theorem 61 vii
Contents viii §5.1 Relative entropy and large deviations for empirical measures 67 . Relative entropy §5.2 §5.3 . . Chapter 5. Chapter 6. §6.1 §6.2 §6.3 Sanov’s theorem Maximum entropy principle Process level large deviations fori.i.d. fields . Setting . Specific relative entropy . Pressure and the large deviationprinciple 67 73 78 83 83 85 91 Part II. Statistical mechanics Formalism for classical lattice systems Chapter 7. §7.1 . Finite-volume model §7.2 . Potentials and Hamiltonians §7.3 §7.4 . . Specifications Phase transition §7.5 §7.6 . . Extreme Gibbs measures Uniqueness for small potentials Chapter 8. 99 99 101 103 108 110 112 Large deviations and equilibrium statistical mechanics 121 §8.1 §8.2 . . Thermodynamic limit of the pressure Entropy and large deviations under Gibbsmeasures 121 124 §8.3 . Dobrushin-Lanford-Ruelle (DLR) variational principle 127 Phase transition in the Ising model Chapter 9. 133 §9.1 §9.2 §9.3 . . . One-dimensional Ising model Phase transition at low temperature Case of no external field 136 138 141 §9.4 . Case of nonzero external field 146 Percolation approach to phase transition 149 Chapter 10. §10.1 §10.2 . Bernoulli bond percolation and random cluster measures 149 . Ising phase transition revisited 153
ix Contents Part III. Additional large deviation topics Chapter 11. Further asymptotics for i.i.d. random variables 161 §11.1 . Refinement of Cramer’s theorem 161 §11.2 . Moderate deviations 164 Large deviations through the limiting generating function 167 Chapter 12. §12.1 §12.2 . . §12.3 . Chapter 13. Essential smoothness and exposed points Gärtner-Ellis theorem Large deviations for the current of particles 167 175 179 Large deviations for Markov chains 187 §13.1 . Relative entropy for kernels 187 §13.2 . Countable Markov chains §13.3 . Finite Markov chains 191 203 Chapter 14. Convexity criterion for large deviations 213 Chapter 15. Nonstationary independent variables 221 §15.1 . Generalization of relative entropy andSanov’s theorem 221 . §15.2 Chapter 16. §16.1 §16.2 . . Proof of the large deviation principle 223 Random walk in a dynamical random environment 233 Quenched large deviation principles Proofs via the Baxter-Jain theorem 234 239 Appendixes Appendix A. Analysis §A.l. Metric spaces and topology §A.2. Measure and integral 259 259 262 §A.3. §A.4. Product spaces Separation theorem 267 268 §A.5. Minimax theorem 269
Contents x Appendix В. Probability §B.l. Independence 273 274 §В.2. §B.3. §B.4. Existence of stochastic processes Conditional expectation Weak topology of probability measures 275 276 §B.5. First limit theorems §B.6. §B.7. Ergodic theory Stochastic ordering 282 282 Appendix C. Inequalities from statistical mechanics §C.l. Griffiths’s inequality §C.2. Griffiths-Hurst-Sherman inequality Appendix D. Nonnegative matrices 278 288 293 293 294 297 Bibliography 299 Notation index 305 Author index 311 General index 313
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| spelling | Rassoul-Agha, Firas 1973- Verfasser (DE-588)1070664553 aut A course on large deviations with an introduction to Gibbs measures Firas Rassoul-Agha ; Timo Seppäläinen Providence, Rhode Island American Mathematical Society [2015] © 2015 xiv, 318 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics volume 162 Literaturverzeichnis Seite 299-303 Large deviations Probabilities Measure theory Gibbs-Maß (DE-588)4157328-6 gnd rswk-swf Große Abweichung (DE-588)4330658-5 gnd rswk-swf Große Abweichung (DE-588)4330658-5 s Gibbs-Maß (DE-588)4157328-6 s DE-604 Seppäläinen, Timo O. 1961- Verfasser (DE-588)1070666106 aut Erscheint auch als Online-Ausgabe 978-1-4704-2222-6 Graduate studies in mathematics volume 162 (DE-604)BV009739289 162 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028021204&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028021204&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rassoul-Agha, Firas 1973- Seppäläinen, Timo O. 1961- A course on large deviations with an introduction to Gibbs measures Graduate studies in mathematics Large deviations Probabilities Measure theory Gibbs-Maß (DE-588)4157328-6 gnd Große Abweichung (DE-588)4330658-5 gnd |
| subject_GND | (DE-588)4157328-6 (DE-588)4330658-5 |
| title | A course on large deviations with an introduction to Gibbs measures |
| title_auth | A course on large deviations with an introduction to Gibbs measures |
| title_exact_search | A course on large deviations with an introduction to Gibbs measures |
| title_full | A course on large deviations with an introduction to Gibbs measures Firas Rassoul-Agha ; Timo Seppäläinen |
| title_fullStr | A course on large deviations with an introduction to Gibbs measures Firas Rassoul-Agha ; Timo Seppäläinen |
| title_full_unstemmed | A course on large deviations with an introduction to Gibbs measures Firas Rassoul-Agha ; Timo Seppäläinen |
| title_short | A course on large deviations with an introduction to Gibbs measures |
| title_sort | a course on large deviations with an introduction to gibbs measures |
| topic | Large deviations Probabilities Measure theory Gibbs-Maß (DE-588)4157328-6 gnd Große Abweichung (DE-588)4330658-5 gnd |
| topic_facet | Large deviations Probabilities Measure theory Gibbs-Maß Große Abweichung |
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| volume_link | (DE-604)BV009739289 |
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