# A Course on Large Deviations With an Introduction to Gibbs Measures

@inproceedings{RassoulAgha2015ACO, title={A Course on Large Deviations With an Introduction to Gibbs Measures}, author={Firas Rassoul-Agha and Timo Sepp{\"a}l{\"a}inen}, year={2015} }

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…

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## References

SHOWING 1-10 OF 72 REFERENCES

Quenched Free Energy and Large Deviations for Random Walks in Random Potentials

- Mathematics
- 2011

We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded…

Large deviation principles for Euclidean functionals and other nearly additive processes

- Mathematics, Computer Science
- 2001

Abstract. We prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or Euclidean space, under approximate additivity and regularity hypotheses. The…

Large Deviations for Empirical Measures of Not Necessarily Irreducible Countable Markov Chains with Arbitrary Initial Measures

- Mathematics
- 2005

All known results on large deviations of occupation measures of Markov processes are based on the assumption of (essential) irreducibility. In this paper we establish the weak* large deviation…

Convexity in the Theory of Lattice Gases

- Physics
- 1979

In this book, Robert Israel considers classical and quantum lattice systems in terms of equilibrium statistical mechanics. He is especially concerned with the characterization of…

Some familiar examples for which the large deviation principle does not hold

- Mathematics
- 1991

For a class of Markov processes (in continuous or discrete time) we show that if the full large deviation holds for normalized occupation time measures Lt(w, ˙) with some rate function J, then the…

Large Deviations and Applications

- Mathematics
- 1984

Large Deviations Cramer's Theorem Multidimensional Version of Cramer's Theorem An Infinite Dimensional Example: Brownian Motion The Ventcel-Freidlin Theory The Exit Problem Empirical Distributions…

The Kardar-Parisi-Zhang Equation and Universality Class

- Mathematics
- 2011

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or…

Probability: Theory and Examples

- Mathematics
- 1990

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a…

Large deviations for empirical measures of Markov chains

- Mathematics
- 1990

In this paper we obtain large-deviation upper and lower bounds for the empirical measure of a Markov chain with general state space, as well as for the associated multivariate empirical measure and…

Gibbs Measures and Phase Transitions

- Physics
- 1988

This comprehensive monograph covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics and serves both as an introductory text and as a reference for the expert.