One-dimensional disordered Ising models by replica and cavity methods
  Using a formalism based on the spectral decomposition of the replicated transfer matrix for disordered Ising models, we obtain several results that apply both to isolated one-dimensional systems and to locally tree-like graph and factor graph (p-spin) ensembles. We present exact analytical expressions, which can be efficiently approximated numerically, for many types of correlation functions and for the average free energies of open and closed finite chains. All the results achieved, with the exception of those involving closed chains, are then rigorously derived without replicas, using a probabilistic approach with the same flavour of cavity method. 
Carlo Lucibello, Flaviano Morone, Tommaso Rizzo
cond-mat.dis-nn cond-mat.stat-mech