From the seminar on Mathematical Statistical Physics in Moscow State University, 1962-1994. Contour technics

摘要

A core problem in the rigorous approach to statistical physics is a mathematical description of phase transitions. In the framework that had been developed at the Moscow seminar (see [Minlos 2012]), this problem emerged as the question of non-uniqueness of the Gibbs distributions for a given potential (or, for a given specification). It was understood at an early stage that different Gibbs distributions can appear in the thermodynamic limit under different boundary conditions. This idea, originated by Peierls in the 1930s [Peierls 1936], was rediscovered in the 1960s by Griffiths [Griffiths 1964; Dobrushin 1965]. The efforts here were aimed at reducing the analysis of Gibbs distributions to studying geometric properties of the so-called contour ensembles. Later, this approach was used by Minlos and Sinai in a detailed analysis of typical configurations of Gibbs random fields in the regime of phase transitions. Further development of these ideas by seminar participants resulted in a description of phase transitions for a wide class of lattice systems.