The phase boundary in dilute and random Ising and Potts ferromagnets

The authors present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple. Their analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary. For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1-p, the critical temperature scales as exp(-J/(kT_c(p))) approximately mod p-p_c mod , implying that the crossover exponent is Phi =1. If the constant couplings are replaced by a distribution F(J) with mass near J=0, quite different crossover behaviour is observed. For example, if F(J) approximately J^alpha then, for p near p_c, T_c(p) approximately mod p-p_c mod ¹ alpha /.