Ising Models on Hyperbolic Graphs II

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph ℋ(v, f), where v is the number of neighbors of each vertex and f is the number of sides of each face. Let Tc be the critical temperature and T′c=sup〈T≤Tc:νf=(ν++ν−)/2〉, where νf is the free boundary condition (b.c.) Gibbs state, ν+ is the plus b.c. Gibbs state and ν− is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., v=f) or if v is sufficiently large (how large depends on f, e.g., v≥35 suffices for any f≥3 and v≥17 suffices for any f≥17) then 0<T′c<Tc, in contrast with that T′c=Tc for Ising models on the hypercubic lattice Zd with d≥2, a result due to Lebowitz.(22) While whenever T<T′c, νf=(ν++ν−)/2. The last result is an improvement in comparison with the analogous statement in refs. 28 and 33, in which it was only proved that νf=(ν++ν−)/2 when T≪T′c and it remains to show in both papers that νf=(ν++ν−)/2 whenever T<T′c. Therefore T′c and Tc divide [0, ∞] into three intervals: [0, T′c), (T′c, Tc), and (Tc, ∞] in which ν+≠ν− but νf=(ν++ν−)/2, ν+≠ν− and νf≠(ν++ν−)/2, and ν+=ν−, respectively.