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Using a formalism based on the spectral decomposition of the replicatedtransfer matrix for disordered Ising models, we obtain several results thatapply both to isolated one-dimensional systems and to locally tree-like graphand factor graph (p-spin) ensembles. We present exact analytical expressions,which can be efficiently approximated numerically, for many types ofcorrelation functions and for the average free energies of open and closedfinite chains. All the results achieved, with the exception of those involvingclosed chains, are then rigorously derived without replicas, using aprobabilistic approach with the same flavour of cavity method.

One-dimensional disordered Ising models by replica and cavity methods