Extremity of the disordered phase in the Ising model on the Bethe lattice (Q910110)

On the Bethe lattice of degree \(k\geq 1\), the Ising model is defined by the Hamiltonian \[ H(\sigma)= -\sum_{<x,y>}J_{xy}\sigma (x)\sigma (y) \] (the sum taken over all pairs of nearest neighbours \(<x,y>\), the spins \(\sigma\) (x) taking the values \(\pm 1)\). In the ferromagnetic Ising model \(J_{xy}\equiv J>0\), whereas in the spin glass model \(J_{xy}\) is random, taking the values \(\pm J\) \((J>0)\) with probability 1/2. Put \(\theta =\tanh (J/T)\). It is known that the critical value for the ferromagnetic Ising model is \(\theta^ F_ c=1/k\) and for the spin glass model \(\theta_ c^{SG}=1/k^{1/2}.\) In the present paper it is shown that for \(0<\theta \leq \theta_ c^{SG}\) the disordered Gibbs distribution is extreme. This settles an open problem which arose e.g. in the paper by \textit{T. Moore} and \textit{J. L. Snell} [J. Appl. Probab. 16, 252-260 (1979; Zbl 0421.60052)].