Ising Models on Hyperbolic Graphs II

Abstract

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 T _c be the critical temperature and T′ _c =sup〈T≤T _c:ν ^f=(ν ⁺+ν ⁻)/2〉, where ν ^f 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 <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 , ν ^f=(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 ν ^f=(ν ⁺+ν ⁻)/2 when T≪T′ _c and it remains to show in both papers that ν ^f=(ν ⁺+ν ⁻)/2 whenever T<T′ _c . Therefore T′ _c and T _c divide [0, ∞] into three intervals: [0, T′ _c ), (T′ _c , T _c), and (T _c, ∞] in which ν ⁺≠ν ⁻ but ν ^f=(ν ⁺+ν ⁻)/2, ν ⁺≠ν ⁻ and ν ^f≠(ν ⁺+ν ⁻)/2, and ν ⁺=ν ⁻, respectively.