PERCOLATION IN THE RANDOM CLUSTER PROCESS AND Q-STATE POTTS-MODEL

The critical probabilities p(T) = p(T) (Q) and p(H) = p(H) (Q) of the random cluster model with parameter Q greater-than-or-equal-to 1 on the square lattice are shown to satisfy p(T) less-than-or-equal-to square-root Q/(1 + square-root Q) less-than-or-equal-to p(H), p(H) less-than-or-equal-to Q(1 - p(T))/(p(T) + Q(1 - p(T))). The last inequality is the Q-analogue of the relation p(T) + p(H) less-than-or-equal-to 1 of ordinary (Q = 1) percolation, though of course in this case equality holds.