Ising models on hyperbolic graphs
The author studies the nearest-neighbour ferromagnetic Ising model on a hyperbolic graph \(G\) that can be thought as a discretisation of the hyperbolic plane. More formally, the graph \(G\) is constructed from a rooted \(k\)-tree \(T_k\) by adding edges connecting some equal-level sites in such a way that at the level \(l\geq 1\) exactly \(k^l-1\) edges are added (as a result, for \(l\geq 1\) there are \(2\) vertices of index \(k+2\) and \(k^l-2\) vertices of index \(k+3\)). It is shown that the Ising model on \(G\) exhibits multiple phase transitions. Namely, let \(\nu^+\), \(\nu^-\), and \(\nu^f\) denote the Gibbs states with plus, minus, and free boundary conditions, respectively, and let \(M\) be the mean of the spin at the origin w.r.t.~\(\nu^+\). Then, for \(k\geq 8\), there exist \(\beta_c\), \(\beta_c'\), and \(\beta_c''\) satisfying \(0<\beta_c<\beta_c'\leq\beta_c''\) such that the following holds: 1) for \(\beta<\beta_c\), \(M=0\) and there is only one Gibbs state; 2) for \(\beta_c<\beta<\beta_c'\), \(M>0\) and \(\nu^f\neq(\nu^++\nu^-)/2\), i.e. there are more than two extremal Gibbs states; 3) for \(\beta>\beta_c''\), \(M>0\) and \(\nu^f=(\nu^++\nu^-)/2\). The proof is based on the Fortuin-Kasteleyn (FK) representation of Ising models and the comparison inequalities between FK and independent percolation measures. It is conjectured that \(\beta_c'=\beta_c''\) and that all the results are valid for \(k\geq 2\).
scientific article; zbMATH DE number 1279710