Pirogov-Sinai theory

• Sergei A. Pirogov*, Institute for Information Transmission Problems, Moscow, Russia
• Milos Zahradnik, Mathematics, Charles University in Prague

Pirogov-Sinai Theory is the rigorous perturbation technique allowing to prove the existence of the phase transition of the 1-st kind. Any symmetry of hamiltonian is not necessarily supposed. The small parameter in many cases (but not always) is exp(-beta). In that sense Pirogov-Sinai theory can be considered as the rigorous version of low-temperature perturbation theory.

The main technical unit of Pirogov-Sinai theory is contour, i.e. the elementary excitation of some ground state of considered system geometrcally surrounding the lakes filled by other ground states. This concept of contour was firstly introduced in the pioneering article by R. Peierls (1936). The mathematical theory of contours was developped in the series of papers by R.A.Minlos and Ya.G.Sinai in 60-s. But in these papers as well as in R. Peierls paper contours were applied only to the models having spin-flip symmetry. Pirogov-Sinai theory appeared in early 70-s as an attempt to abandon this restriction. Pirogov-Sinai theory was then developped in the papers by Ya. G. Sinai - E. I. Dinaburg -A. E. Mazel, M. Zahradnik, R. Kotecky and others. E. Presutti et al.(2000) generalized this theory (formulated initially for lattice systems) to some class of continious-space systems.

See also

Kolmogorov-Sinai entropy

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