Abstract

We consider a one-dimensional spin model with the long-range random Hamiltonian given by H[σ]=-12∑x≠yJx,yσxσy|x-y|α0+αx,y . The randomness is considered in both the pairwise interaction Jx,y and in its decaying parameter with slowest value α plus a non-negative random variable αx,y . We prove the loss of stability at α= 1 / 2 . We also prove the existence of the free energy at the thermodynamic limit when α> 1 / 2 . Furthermore, we show uniqueness of the equilibrium state for α> 3 / 2 in the strong sense.