Monte Carlo Methods in Statistical Mechanics: Foundations and ...

Finally, let us see what happens if we sumover the f g at xed fng. Each occupied
bond hiji imposes a constraint that the spins i and j mustbeinthe same state,
but otherwise the spins are unconstrained. We therefore group the sites into connected
clusters (two sites are in the same cluster if they can be joined by apath of occupied
bonds) then all the spins within a cluster mustbeinthe samestate (all q values are
equally probable), and distinct clusters are independent. It follows that
Z = X
fng
0
@ Y
hiji: nij=1
pij
1 0
A @ Y
hiji: nij=0
1
(1 ; pij) A q C(n) (6.7)
where C(n) isthe number of connected clusters (including one-siteclusters) in the graph
whose edges are the bonds having nij =1. The corresponding probability distribution,
RC(n) = Z ;1
0
@ Y
hiji: nij=1
pij
1 0
A @ Y
hiji: nij=0
1
(1 ; pij) A q C(n) (6.8)
is called the random-cluster model with parameter q. This is a generalized bondpercolation
model, with non-local correlations coming fromthe factor q C(n) forq =1
it reduces to ordinary bond percolation. Note, by the way, that inthe random-cluster
model (unlike the Potts andFKSWmodels), q is merely a parameter it can take any
positiverealvalue, not just 2 3:::.Sotherandom-cluster model de nes, in some sense,
an analytic continuation of the Potts model to non-integer q ordinary bond percolation
corresponds to the \one-state Potts model".
We have already veri ed the following factsabout the FKSWmodel:
a) ZP otts = ZFKSW = ZRC.
b) The marginal distribution of FKSW on the Potts variables f g (integrating out
the fng) is precisely the Potts model Potts( ).
c) The marginal distribution of FKSW on the bondoccupation variables fng (integrating
outthe f g) is precisely the random-cluster model RC(n).
The conditional distributions of FKSW are also simple:
d) The conditional distribution of the fng given the f g is as follows: independently
for each bond hiji, onesets nij = 0 in case i 6= j, andsets nij =0 1with
probability 1; pijpij, respectively,incase i = j.
e) The conditional distribution of the f g given the fng is as follows: independently
for each connected cluster, one sets all the spins i in the cluster to thesamevalue,
chosen equiprobably from f1 2:::qg.
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