Blume-Capel Model, remarks
Representation of the Blume-Capel Model,
G. R. Grimmett and B. T. Graham,
Journal of Statistical Physics, 2006.
- 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
the author's webpage. In particular, the statement after equation (5.1)
is incorrect, but the gap is covered by the
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.