Limit Theorems and Some Applications in Statistical Physics

1. Preliminary Probabilistic Results.- 1.1 Convergence of Probability Measures.- 1.2 Central Limit Theorem for Sums of Random Vectors.- 1.3 Invariance Principle.- 1.4 Stationary Random Processes and Fields on the Lattice.- 1.5 Slowly Varying Functions.- 1.6 Some Probabilistic Inequalities.- 1.7 Uniformly Integrable Sequences.- 2. Weak Dependence Conditions for Random Processes and Fields.- 2.1 Measures of Dependence.- 2.2 Weak Dependence Conditions for Random Processes. Classical Examples.- 2.3 Davydov's Examples.- 2.4 Herrndorf's Examples.- 2.5 Gaussian Random Sequences.- 2.6 Weak Dependence Conditions for Random Fields. Dobrushin's Example.- 2.7 Additions.- 3. Asymptotic Behavior of the Variance, Estimates on the Moments and Some Probabilistic Inequalities for Sums of Weakly Dependent Random Variables.- 3.1 The Variance of the Sum of Random Variables.- 3.2 Estimates on Moments of Sums of Weakly Dependent Random Variables.- 3.3 Some Probabilistic Inequalities.- 4. Methods.- 4.1 Bernstein's Method.- 4.2 Gordin's Method.- 4.3 Stein's Method.- 4.4 Moments' (Semi-Invariants') Method.- 5. Limit Theorems for Random Processes.- 5.1 Central Limit Theorem for ?-mixing Stationary Random Processes.- 5.2 Rate of Convergence in Central Limit Theorem for ?-mixing Stationary Random Processes.- 5.3 Invariance Principle for Stationary Random Processes with ?-mixing Condition.- 5.4 Law of Iterated Logarithm for Stationary Random Processes Satisfying the ?-mixing Condition.- 5.5 Limit Theorems for ?-mixing Stationary Random Processes.- 5.6 Limit Theorems for Stationary Random Processes Satisfying the ?-mixing Condition.- 5.7 Limit Theorems for ?-mixing Stationary Random Processes.- 6. Limit Theorems under Generalized Mixing Conditions.- 6.1 Distances in Space of Probability Measures, Kantorovioh-Vasershtein Metric.- 6.2 Generalized Mixing Conditions for Random Processes.- 6.3 Central Limit Theorem.- 6.4 Limit Theorems of Non-Commutative Theory of Probability.- 7. Limit Theorems for Random Fields.- 7.1 Sequences of Sets Tending to Infinity.- 7.2 Central Limit Theorem for Random Fields.- 7.3 Rate of Convergence in Central Limit Theorem for Random Fields.- 7.4 Law of Iterated Logarithm for Random Fields.- 8. Description of Random Fields by Means of Conditional Probability.- 8.1 Existence of Random Fields with Given Conditional Distribution.- 8.2 Uniqueness of Random Fields with Given Conditional Distribution.- 8.3 Decay of Correlation and Central Limit Theorem.- 9. Gibbs Random Fields.- 9.1 Existence and Uniqueness of Gibbs Random Fields, Weak Dependence of Components.- 9.2 Thermodynamical Limit, Existence of Free Energy.- 9.3 Strong Convexity of Free Energy, Linear Growth of Variance of Energy.- 9.4 Limit Theorems for Gibbs Random Fields.- 9.5 Cluster Properties and Mixing Conditions for Gibbs Random Fields with Vacuum Potential.- 9.6 Additions.- Some Additional Remarks.- References.