Thermodynamic scaling laws for a dilute ferromagnet in the percolation limit by series methods

The nth field derivative of the free energy of a randomly dilute Ising model in the percolation limit J>>k B t is known to be determined by the nth moment m n (p) of the cluster size distribution. Analytic results for the Bethe lattice and series expansions for two- and three-dimensional lattices are found to support the 'constant gap' hypothesis. The scaling form for the singular part of the free energy is given. In two and three dimensions the series expansions have been analysed for p<p c only, where the critical probability p c is the concentration of magnetic atoms required for a ferromagnetic transition.