### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 893-904 |

Number of pages | 12 |

Journal | Journal of Statistical Physics |

Volume | 100 |

Issue number | 5-6 |

State | Published - Sep 1 2000 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*100*(5-6), 893-904.