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 _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if (Tin(T_{c,sqrt{k}}, T_{c,k_{0}})), with (sqrt{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 ₀≠k′₀. Therefore if (Tin (T_{c,sqrt{k}}, T_{c,sqrt{k}+1})), (T_{c,sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}(cup(bigcup_{k_{0}:sqrt{k}.