Blume-Capel Model, remarks

Remarks on Random-Cluster Representation of the Blume-Capel Model,

G. R. Grimmett and B. T. Graham,

- In Theorem 5.9, the lower case phi should be upper case.
- In line 7 after (7.14), the lower case phi should be upper case.
- Use has been made of Theorem 2.3 of Ref 28, and this is unfortunately incorrect. See the author's webpage. In particular, the statement after equation (5.1) is incorrect, but the gap is covered by the revised version of Theorem 2.3, and the claims in the current paper remain valid.
- The proof of Theorem 6.1(b) is sketchy, and is better explained as follows. Since a_1 = 1, \Upsilon_1 is simply the random-cluster measure on G. \Upsilon_2 is a mixture of random-cluster measures on subsets V_\psi of V, with p_2 \le p_1 and q_2 \ge q_1. Each of these measures is stochastically dominated by \Upsilon_1, and therefore so is the mixture \Upsilon_2.