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We calculate equilibrium solutions for Ising spin models on `small world'lattices, which are constructed by super-imposing random and sparse Poissoniangraphs with finite average connectivity c onto a one-dimensional ring. Thenearest neighbour bonds along the ring are ferromagnetic, whereas thosecorresponding to the Poisonnian graph are allowed to be random. Our models thusgenerally contain quenched connectivity and bond disorder. Within the replicaformalism, calculating the disorder-averaged free energy requires thediagonalization of replicated transfer matrices. In addition to developing thegeneral replica symmetric theory, we derive phase diagrams and calculateeffective field distributions for two specific cases: that of uniform sparselong-range bonds (i.e. `small world' magnets), and that of (+J/-J) randomsparse long-range bonds (i.e. `small world' spin-glasses).Comment: 22 pages, LaTeX, IOP macros, eps figure

Replicated transfer matrix analysis of Ising spin models on small world lattices