Rassoul-agha F., Seppelainen T. A Course on Large Deviations

Salt Lake City: Department of Mathematics, 2014. - 329p.Part I covers core general large deviation theory, the relevant convex analysis, and the large deviations of i.i.d. processes on three levels: Cram er's theorem, Sanov's theorem, and the process level LDP for i.i.d. variables indexed by a multidimensional square lattice. Part II introduces Gibbs measures and proves the Dobrushin-Lanford-Ruelle variational principle that characterizes translation-invariant Gibbs measures. After this we study the phase transition of the Ising model.
Part II ends with a chapter on the Fortuin-Kasteleyn random cluster model and the percolation approach to Ising phase transition.
Part III develops the large deviation themes of Part I in several directions. Large deviations of i.i.d. variables are complemented with moderate deviations and with more precise large deviation asymptotics. The Gartner-Ellis theorem is developed carefully, together with the necessary additional convex analysis beyond the basics covered in Part I. From large deviations of i.i.d. processes we move on to Markov chains, to nonstationary independent random variables, and nally to random walk in a dynamical random environment. The last two topics give us an opportunity to apply another approach to proving large deviation principles, namely the Baxter-Jain theorem. The Baxter-Jain theorem has not previously appeared in textbooks, and its application to random walk in random environment is new.