Low-temperature studies of random ising models

A brief discussion is given of the application of the methods of low-temperature or excitation expansions to random Iaing models. A specific investigation is then undertaken of the low-temperature behaviour of the random-field model A relationship is established between the δ-function distribution with equal probabilities of positive and negative fixed fields H₀and a site percolation process on the lattice with p= ½. At T = 0 successive percolation clusters give rise to first-order transitions at different values of H₀ There is a significant difference in behaviour between lattices for which p_c < ½ which have an infinite cluster and p_c > ½ which do not. For standard lattices, no matter how small is H₀ there are overturned clusters, but for the Bethe lattice there is a range of H₀ for which the ground state is one of ferromagnetic order. By allowing the coordination number to become large in the latter system, the results of the mean-field approximation are reproduced. The above considerations do not apply to a Gaussian distribution of fields, and the absence of a first-order transition can then be understood. Since large clusters overturn for small fields, there are clear indications of metastable behaviour.