Journal of Statistical Physics

, Volume 100, Issue 5–6, pp 893–904 | Cite as

Ising Models on Hyperbolic Graphs II

  • C. Chris Wu

Abstract

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph ℋ(vf), 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 Tc=sup〈TTc: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<Tc<Tc, in contrast with that Tc=Tc for Ising models on the hypercubic lattice Zd with d≥2, a result due to Lebowitz.(22) While whenever T<Tc, 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 TTc and it remains to show in both papers that νf=(ν++ν)/2 whenever T<Tc. Therefore Tc and Tc divide [0, ∞] into three intervals: [0, Tc), (TcTc), and (Tc, ∞] in which ν+ν but νf=(ν++ν)/2, ν+ν and νf≠(ν++ν)/2, and ν+=ν, respectively.

Ising models percolation hyperbolic graphs 

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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • C. Chris Wu
    • 1
  1. 1.Department of MathematicsPenn State University, Beaver CampusMonaca

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