03-08E

# 03-08E

Mathematical theory of the Ising model and its generalizations: an introduction

An introduction into the rigorous theory of equilibrium states of a number of lattice models of classical and quantum statistical physics is given. Generalized Ising models with discrete, continuous, bounded and unbounded spins, translation invariant and with a hierarchical structure; quantum spin models, models of interacting quantum anharmonic oscillators are considered. For the classical models, certain properties of local Gibbs states, such as the Lee-Yang theorem, correlation inequalities, phase transitions, self-similarity, are discussed. For the quantum models, an approach based on functional integration is presented on an introductory level.