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

d = 2 Ising Model
L intE
64 1575 ( 10) 5.25 (0.30)
128 5352 ( 53) 7.05 (0.67)
256 17921 (109) 6.83 (0.40)
512 59504 (632) 7.99 (0.81)
Table 1: Susceptibility andautocorrelation time intE (E =energy slowest mode) for
two-dimensional Ising model at criticality, usingSwendsen-Wang algorithm. Standard
error is shown in parentheses.
nonzero FKSW-measure is accessible from every other. So the SW algorithm is at
least a correct algorithm for simulating the FKSWmodel. It is also an algorithm for
simulatingthe Potts and random-cluster models, since expectations in these two models
are equal to the corresponding expectations in the FKSWmodel.
Historical remark. The random-cluster model was introduced in 1969 by Fortuin
and Kasteleyn [58] they derived the identity ZPotts = ZRC, alongwiththe correlationfunction
identity (6.9) and some generalizations. These relations were rediscovered
several times during the subsequent two decades [59]. Surprisingly, however, no one
seems to have noticed the joint probability distribution FKSW that underlay all these
identities this was discovered implicitly by Swendsen and Wang [27], and was made
explicit by Edwards and Sokal [60].
It is certainly plausible that the SW algorithm might have lesscritical slowing-down
than the conventional (single-spin-update) algorithms: the reasonisthat a local movein
one setofvariables can have highly nonlocal e ects inthe other. For example, setting
nb =0onasingle bond may disconnect a cluster, causing abigsubset of the spins in
that cluster to be ipped simultaneously. In some sense,therefore, the SW algorithm
is a collective-mode algorithm in which the collective modes are chosen by the system
rather than imposed from the outside asinmulti-grid. (The miracle is that thisisdone
in a way that preserves the correct Gibbs measure.)
How well does the SW algorithm perform? In at least some cases,the performance
is nothing short of extraordinary. Table 1 shows some preliminary data [61]onatwodimensional
Ising model at the bulkcritical temperature. These data are consistent
with the estimate SW L 0:35 [27]. 25 By contrast, the conventional single-spin- ip
25 But precisely because rises so slowly with L, good estimates of the dynamic critical exponent
will require the useofextremely large lattices. Even with lattices up to L =512,weareunable to
distinguish convincingly between z 0:35 and z 0. Note Added 1996: For more recent and
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