Large deviations theory

Abstract: The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

Abstract: I: Large Deviations and Statistical Mechanics.- I. Introduction to Large Deviations.- I.1. Overview.- I.2. Large Deviations for I.I.D. Random Variables with a Finite State Space.- I.3. Levels-1 and 2 for Coin Tossing.- I.4. Levels-1 and 2 for I.I.D. Random Variables with a Finite State Space.- I.S. Level-3: Empirical Pair Measure.- I.6. Level-3: Empirical Process.- I.7. Notes.- I.B. Problems.- II. Large Deviation Property and Asymptotics of Integrals.- II.1. Introduction.- II.2. Levels-1, 2, and 3 Large Deviations for I.I.D. Random Vectors.- II.3. The Definition of Large Deviation Property.- II.4. Statement of Large Deviation Properties for Levels-1, 2, and 3.- II.5. Contraction Principles.- II.6. Large Deviation Property for Random Vectors and Exponential Convergence.- II.7. Varadhan's Theorem on the Asymptotics of Integrals.- II.8. Notes.- II.9. Problems.- III. Large Deviations and the Discrete Ideal Gas.- III.1. Introduction.- III.2. Physics Prelude: Thermodynamics.- III.3. The Discrete Ideal Gas and the Microcanonical Ensemble.- III.4. Thermodynamic Limit, Exponential Convergence, and Equilibrium Values.- III.5. The Maxwell-Boltzmann Distribution and Temperature.- III.6. The Canonical Ensemble and Its Equivalence with the Microcanonical Ensemble.- III.7. A Derivation of a Thermodynamic Equation.- III.8. The Gibbs Variational Formula and Principle.- III.9. Notes.- III.10. Problems.- IV. Ferromagnetic Models on ?.- IV.1. Introduction.- IV.2. An Overview of Ferromagnetic Models.- IV.3. Finite-Volume Gibbs States on ?.- IV.4. Spontaneous Magnetization for the Curie-Weiss Model.- IV.5. Spontaneous Magnetization for General Ferromagnets on ?.- IV.6. Infinite-Volume Gibbs States and Phase Transitions.- IV.7. The Gibbs Variational Formula and Principle.- IV.8. Notes.- IV.9. Problems.- V. Magnetic Models on ?D and on the Circle.- V.1. Introduction.- V.2. Finite-Volume Gibbs States on ?D, D ? 1.- V.3. Moment Inequalities.- V.4. Properties of the Magnetization and the Gibbs Free Energy.- V.5. Spontaneous Magnetization on ?D, D ? 2, Via the Peierls Argument.- V.6. Infinite-Volume Gibbs States and Phase Transitions.- V.7. Infinite-Volume Gibbs States and the Central Limit Theorem.- V.8. Critical Phenomena and the Breakdown of the Central Limit Theorem.- V.9. Three Faces of the Curie-Weiss Model.- V.10. The Circle Model and Random Waves.- V.11. A Postscript on Magnetic Models.- V.12. Notes.- V.13. Problems.- II: Convexity and Proofs of Large Deviation Theorems.- VI. Convex Functions and the Legendre-Fenchel Transform.- VI.1. Introduction.- VI.2. Basic Definitions.- VI.3. Properties of Convex Functions.- VI.4. A One-Dimensional Example of the Legendre-Fenchel Transform.- VI.5. The Legendre-Fenchel Transform for Convex Functions on ?d.- VI.6. Notes.- VI.7. Problems.- VII. Large Deviations for Random Vectors.- VII.1. Statement of Results.- VII.2. Properties of IW.- VII.3. Proof of the Large Deviation Bounds for d = 1.- VII.4. Proof of the Large Deviation Bounds for d ? 1.- VII.5. Level-1 Large Deviations for I.I.D. Random Vectors.- VII.6. Exponential Convergence and Proof of Theorem II.6.3.- VII.7. Notes.- VII.8. Problems.- VIII. Level-2 Large Deviations for I.I.D. Random Vectors.- VIII.1. Introduction.- VIII.2. The Level-2 Large Deviation Theorem.- VIII.3. The Contraction Principle Relating Levels-1 and 2 (d = 1).- VIII.4. The Contraction Principle Relating Levels-1 and 2 (d ? 2).- VIII.5. Notes.- VIII.6. Problems.- IX. Level-3 Large Deviations for I.I.D. Random Vectors.- IX.1. Statement of Results.- IX.2. Properties of the Level-3 Entropy Function.- IX.3. Contraction Principles.- IX.4. Proof of the Level-3 Large Deviation Bounds.- IX.5. Notes.- IX.6. Problems.- Appendices.- Appendix A: Probability.- A.1. Introduction.- A.2. Measurability.- A.3. Product Spaces.- A.4. Probability Measures and Expectation.- A.S. Convergence of Random Vectors.- A.6. Conditional Expectation, Conditional Probability, and Regular Conditional Distribution.- A.7. The Kolmogorov Existence Theorem.- A.8. Weak Convergence of Probability Measures on a Metric Space.- Appendix B: Proofs of Two Theorems in Section II.7.- B.1. Proof of Theorem II.7.1.- B.2. Proof of Theorem II.7.2.- Appendix C: Equivalent Notions of Infinite-Volume Measures for Spin Systems.- C.1. Introduction.- C.2. Two-Body Interactions and Infinite-Volume Gibbs States.- C.3. Many-Body Interactions and Infinite-Volume Gibbs States.- C.4. DLR States.- C.5. The Gibbs Variational Formula and Principle.- C.6. Solution of the Gibbs Variational Formula for Finite-Range Interactions on ?.- Appendix D: Existence of the Specific Gibbs Free Energy.- D.1. Existence Along Hypercubes.- D.2. An Extension.- List of Frequently Used Symbols.- References.- Author Index.

Abstract: 1. An Introductory Example: Independent Random Walks.- 2. Some Interacting Particle Systems.- 3. Weak Formulations of Local Equilibrium.- 4. Hydrodynamic Equation of Symmetric Simple Exclusion Processes.- 5. An Example of Reversible Gradient System: Symmetric Zero Range Processes.- 6. The Relative Entropy Method.- 7. Hydrodynamic Limit of Reversible Nongradient Systems.- 8. Hydrodynamic Limit of Asymmetric Attractive Processes.- 9. Conservation of Local Equilibrium for Attractive Systems.- 10. Large Deviations from the Hydrodynamic Limit.- 11. Equilibrium Fluctuations of Reversible Dynamics.- Appendices.- 1. Markov Chains on a Countable Space.- 1.1 Discrete Time Markov Chains.- 1.2 Continuous Time Markov Chains.- 1.3 Kolmogorov's Equations, Generators.- 1.4 Invariant Measures, Reversibility and Adjoint Processes.- 1.5 Some Martingales in the Context of Markov Processes.- 1.6 Estimates on the Variance of Additive Functionals of Markov Processes.- 1.7 The Feynman-Kac Formula.- 1.8 Relative Entropy.- 1.9 Entropy and Markov Processes.- 1.10 Dirichlet Form.- 1.11 A Maximal Inequality for Reversible Markov Processes.- 2. The Equivalence of Ensembles, Large Deviation Tools and Weak Solutions of Quasi-Linear Differential Equations.- 2.1 Local Central Limit Theorem and Equivalence of Ensembles.- 2.2 On the Local Central Limit Theorem.- 2.3 Remarks on Large Deviations.- 2.4 Weak Solutions of Nonlinear Parabolic Equations.- 2.5 Entropy Solutions of Quasi-Linear Hyperbolic Equations.- 3. Nongradient Tools: Spectral Gap and Closed Forms.- 3.1 On the Spectrum of Reversible Markov Processes.- 3.2 Spectral Gap for Generalized Exclusion Processes.- 3.4 Closed and Exact Forms.- 3.5 Comments and References.- References.