# Sannolikhet, statistik och kombinatorik: Phase transition in long-range Ising models and uniqueness of equilibrium states

- Date:
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
- Lecturer: Anders Öberg
- Contact person: Xing Shi Cai
- Seminarium

Abstract: In the talk I will give some background to the problem of representing the equilibrium states in dynamical systems theory for two-sided long-range Ising models as invariant measures using transfer operator techniques. This was pioneered by Ruelle, Sinai and Bowen. They used Hölder continuous potentials, however, and derived a translation by means of a Hölder continuous transfer function that becomes an eigenfunction of the transfer operator, viewed in the one-sided dynamical systems context. Somewhat later Peter Walters proved that if a one-sided potential has summable variations, then there exists a continuous eigenfunction, but no one has provided weaker conditions for a continuous eigenfuntion. Since there are now weaker conditions for the uniqueness of Gibbs states in general, one would like to think that such conditions would provide a continuous transfer function or eigenfunction. We prove that for the important class of Dyson potentials, there exists a continuous eigenfunction for the transfer operator arbitrary close to the inverse critical temperature at the critical level. In the proof we use the random cluster model. This is joint work with Anders Johansson.