On the Purity of the Limiting Gibbs State for the Ising Model on the Bethe Lattice

1. The quest ion of the possibi l i ty of the d isordered l imiting Gibbs state (i.e., with zero effective field) of the ferromagnetic Ising model on the Bethe latt ice to be pure below the ferromagnetic critical tempera ture Tc has been formula ted as an open p rob lem in refs. 2 and 3 (some nonhomogeneous pure Gibbs states were const ructed in refs. 4 and 5). In ref. 1 it was proved that the d isordered phase is pure up to the spin-glass critical temperasc ture Tc , but as was found out later in correspondence between the au thor of ref. 1 and Hans -Ot to Georgi i , in the recursive formula used in ref. 1 for the p robabi l i ty measures vx [see the formula (3.7) be low] , the mult ipl ier a(hv) was missed both in numera to r and denominator . A corrected version of the p roo f tr) was based on an inequali ty valid for the Bethe latt ice z* of degree k ~< 6. In the present note we give a new p roo f valid for all k.