Random-cluster representation of the Blume-Capel model

The so-called diluted-random-cluster model may be viewed as a random-cluster representation of the Blume–Capel model. It has three parameters, a vertex parameter $a$, an edge parameter $p$, and a cluster weighting factor $q$. Stochastic comparisons of measures are developed for the vertex marginal' when $q\in[1,2]$, and the edge marginal’ when $q\in [1,\text{oo})$. Taken in conjunction with arguments used earlier for the random-cluster model, these permit a rigorous study of part of the phase diagram of the Blume–Capel model.