SOME FACTS ABOUT THE MATHEMATICAL THEORY OF THE ISING MODEL AND ITS GENERALIZATIONS

Yuri Kozitsky

The first part of the lecture gives an outlook of the main aspects of the mathematical theory of the Ising model. The existance and differentiability of the infinite volume free energy density, including the properties connected with the Lee-Yang theorem, are discussed. Then the equilibrium state of the model as a probability measure on the space of configurations is introduced, a number of its properties are described. In particular, the nonuniqueness/ phase transitions properties are discussed on the base of Dobrushin’s criterium, as well as of the Lebowitz/Martin-Loef analitidty results. In the second part of the lectm-e, the above scheme is applied to the Ising model with a transverse field (De Gennes model), which contains non-comutative operators. Here the Euclidean approach, hi which quantum states are represented by probability measures, is employed.