Ising models on hyperbolic graphs II

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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 Tc be the critical temperature and T′c = sup{ T ≤ Tc : vf = (v+ + v-)/2}, where vf 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 < 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 vf = (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 vf = (v+ + v-)/2 when T ≪ T′c and it remains to show in both papers that vf = (v+ + v-)/2 whenever T < T′c. Therefore T′c and Tc divide [0, cursive Greek chi] into three intervals: [0, T′c), (T′c,Tc), and (Tc, cursive Greek shi] in which v+ ≠ v-, but vf = (v+ + v-)/2, v+ ≠ v- and vf ≠ (v+ + v-)/2, and v+ = v-, respectively.

Original languageEnglish (US)
Pages (from-to)893-904
Number of pages12
JournalJournal of Statistical Physics
Issue number5-6
StatePublished - Sep 1 2000

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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