A Course on Large Deviations With an Introduction to Gibbs Measures

Large deviations: General theory and i.i.d. processes Introductory discussion The large deviation principle Large deviations and asymptotics of integrals Convex analysis in large deviation theory Relative entropy and large deviations for empirical measures Process level large deviations for i.i.d. fields Statistical mechanics Formalism for classical lattice systems Large deviations and equilibrium statistical mechanics Phase transition in the Ising model Percolation approach to phase transition Additional large deviation topics Further asymptotics for i.i.d. random variables Large deviations through the limiting generating function Large deviations for Markov chains Convexity criterion for large deviations Nonstationary independent variables Random walk in a dynamical random environment Appendixes: Analysis Probability Inequalities from statistical mechanics Nonnegative matrices Bibliography Notation index Author index General index

[1]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[2]  T. Liggett.  Interacting Particle Systems , 1985 .

[3]  Robert B. Griffiths,et al.  Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet , 1964 .

[4]  R. Ellis,et al.  LARGE DEVIATIONS FOR A GENERAL-CLASS OF RANDOM VECTORS , 1984 .

[5]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[6]  Hans Föllmer,et al.  Random fields and diffusion processes , 1988 .

[7]  N. Jain,et al.  Large Deviation Lower Bounds for Additive Functionals of Markov Processes , 1990 .

[8]  T. Seppäläinen,et al.  Quenched Free Energy and Large Deviations for Random Walks in Random Potentials , 2011, 1104.3110.

[9]  Gábor Kassay.  A simple proof for König's minimax theorem , 1994 .

[10]  H. König.  Über das von Neumannsche Minimax-Theorem , 1968 .

[11]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[12]  D. Huse,et al.  Pinning and roughening of domain walls in Ising systems due to random impurities. , 1985, Physical review letters.

[13]  A. Messager,et al.  Correlation inequalities and uniqueness of the equilibrium state for the plane rotator ferromagnetic model , 1978 .

[14]  David Ruelle,et al.  Observables at infinity and states with short range correlations in statistical mechanics , 1969 .

[15]  Rick Durrett,et al.  Probability: Theory and Examples, 4th Edition , 2010 .

[16]  Th. Riedrich.  R. G. Bartle, The Elements of Real Analysis. XIV + 447 S. m. Fig. New York/London/Sydney 1964. John Wiley & Sons, Inc. Preis geb. 83 s. net , 1965 .

[17]  Wilhelm Schlag,et al.  A Course in Complex Analysis and Riemann Surfaces , 2014 .

[18]  Ofer Zeitouni,et al.  Random walks in random environments , 2003, math/0304374.

[19]  R. Israel.  Convexity in the Theory of Lattice Gases , 1979 .

[20]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[21]  Olle Häggström,et al.  Random-cluster representations in the study of phase transitions , 1996 .

[22]  Liming Wu,et al.  Large Deviations for Empirical Measures of Not Necessarily Irreducible Countable Markov Chains with Arbitrary Initial Measures , 2005 .

[23]  Instability of phase coexistence and translation invariance in two dimensions , 1979 .

[24]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

[25]  H. Beijeren.  Interface sharpness in the Ising system , 1975 .

[26]  J. Bricmont.  The statistical mechanics of lattice gases , 1996 .

[27]  J. Yukich,et al.  Large deviation principles for Euclidean functionals and other nearly additive processes , 2001 .

[28]  J. Fröhlich,et al.  Kosterlitz-Thouless Transition in the Two-Dimensional Plane Rotator and Coulomb Gas , 1981 .

[29]  J. Fröhlich,et al.  Spin waves, vortices, and the structure of equilibrium states in the classicalXY model , 1983 .

[30]  J. R. Baxter,et al.  Some familiar examples for which the large deviation principle does not hold , 1991 .

[31]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[32]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[33]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[34]  A. Acosta.  Large deviations for empirical measures of Markov chains , 1990 .

[35]  S. Resnick.  Adventures in stochastic processes , 1992 .

[36]  S. Varadhan.  Large Deviations and Applications , 1984 .

[37]  R. Dobrushin.  Existence of a Phase Transition in Two-Dimensional and Three-Dimensional Ising Models , 1965 .

[38]  Klaus Fritzsche,et al.  From holomorphic functions to complex manifolds , 2002 .

[39]  R. Durrett.  Probability: Theory and Examples , 1993 .

[40]  L. J. Landau,et al.  On the uniqueness of the equilibrium state for plane rotators , 1977 .

[41]  Yvonne Freeh,et al.  Non Negative Matrices And Markov Chains Springer Series In Statistics , 2016 .

[42]  THE KARDAR-PARISI-ZHANG,et al.  The Kardar-Parisi-Zhang Equation and Universality Class , 2011 .

[43]  Michael S. Waterman,et al.  An extreme value theory for long head runs , 1986 .

[44]  Erwin Schrödinger.  Statistical thermodynamics : a course of seminar lectures delivered in January-March 1944, at the School of Theoretical Physics, Dublin Institute for Advanced Studies , 1957 .

[45]  V. Strassen.  The Existence of Probability Measures with Given Marginals , 1965 .

[46]  Rohini Kumar.  Space-Time Current Process for Independent Ran- dom Walks in One Dimension , 2008, 0807.3313.

[47]  S. Varadhan,et al.  Large deviations , 2019, Graduate Studies in Mathematics.

[48]  Gerald B. Folland,et al.  Real Analysis: Modern Techniques and Their Applications , 1984 .

[49]  I. Dinwoodie.  A Note on the Upper Bound for I.I.D. Large Deviations , 1991 .

[50]  S. Varadhan,et al.  Probability Theory , 2001 .

[51]  David A. Huffman,et al.  A method for the construction of minimum-redundancy codes , 1952, Proceedings of the IRE.

[52]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .