Phase Transitions for a model with uncountable set of spin values on a Cayley tree

Abstract

In this paper we consider a model with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\ge 2$. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For $k=2$ and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e. there are phase transitions).

10 pages. arXiv admin note: substantial text overlap with arXiv:1202.2542, arXiv:1202.1722