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 r is the number of neighbors of each vertex and f is the number of sides of each face. Let T_c be the critical temperature and T′_c = sup{ T ≤ T_c : v^f = (v⁺ + v^-)/2}, where v^f is the free boundary condition (b.c.) Gibbs state, v⁺ is the plus b.c. Gibbs state and v^- is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., r = f) or if v is sufficiently large (how large depends on f, e.g., r ≥ 35 suffices for any f ≥ 3 and v ≥ 17 suffices for any f ≥ 17) then 0 < T′_c < T_c, in contrast with that T′_c = T_c for Ising models on the hypercubic lattice Z^d with d ≥ 2, a result due to Lebowitz.⁽²²⁾ While whenever T < T′_c v^f = (v⁺ + v^-)/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 v^f = (v⁺ + v^-)/2 when T ≪ T′_c and it remains to show in both papers that v^f = (v⁺ + v^-)/2 whenever T < T′_c. Therefore T′_c and T_c divide [0, cursive Greek chi] into three intervals: [0, T′_c), (T′_c,T_c), and (T_c, cursive Greek shi] in which v⁺ ≠ v^-, but v^f = (v⁺ + v^-)/2, v⁺ ≠ v^- and v^f ≠ (v⁺ + v^-)/2, and v⁺ = v^-, respectively.