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

摘要

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 c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if T Î (T_c,Ök, T_c,k0)Tin(T_{c,sqrt{k}}, T_{c,k_{0}}), with Ök < k₀ < ksqrt{k}k,k₀{mathcal{G}}_{k,k_{0}} of Gibbs measures. Moreover G_k,k0 ¹ G_k,k¢0{mathcal{G}}_{k,k_{0}}ne {mathcal{G}}_{k,k'_{0}}, for k ₀≠k′₀. Therefore if T Î (T_c,Ök, T_c,Ök+1)Tin (T_{c,sqrt{k}}, T_{c,sqrt{k}+1}), T_c,Ök+1 < T_c,kT_{c,sqrt{k}+1}k₀:Ök < k₀ < kG_k,k0)cup(bigcup_{k_{0}:sqrt{k}