Random spin systems: Some rigorous results

Several general results are obtained for a system of spins on a lattice in which the various lattice sites are occupied at random, and the spins, if present, interact via a general Heisenberg or Ising interaction decreasing sufficiently rapidly with distance. It is shown that the free energy per site exists in the limit of an infinite system, is a continuous function of concentration, and has the usual convexity (stability) properties. For Ising systems with interactions of finite range, the free energy is an analytic function of concentration and magnetic field for a suitable range of these variables. The random Ising ferromagnet on a square lattice (or simple cubic lattice) with nearest-neighbor interactions is shown to exhibit a spontaneous magnetization at sufficiently high concentrations and low temperatures.