A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree.

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥ T= J/arctan (1/ k) the limiting Gibbs measure is unique, and for T< T there are uncountably many extreme Gibbs measures. In the Letter we show that if $T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})$, with $\sqrt{k}<k_{0}<k$ then there is a new uncountable set ${\mathcal{G}}_{k,k_{0}}$ of Gibbs measures. Moreover ${\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}$, for k≠ k′. Therefore if $T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})$, $T_{c,\sqrt{k}+1}<T_{c,k}$ then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures} $\cup(\bigcup_{k_{0}:\sqrt{k}<k_{0}<k}{\mathcal{G}}_{k,k_{0}})$. [ABSTRACT FROM AUTHOR]