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258 Order and disorder in spin systems

Fig. 5.25 Large cluster with 1548 up spins in a 64× 64 Ising model with
periodic boundary conditions (from Alg. 5.9 (cluster-ising), β =0.43).

makes us understand the great potential payoff frominvestments in algo-
rithm design.The implementation ofcluster algorithms such as Alg.5.9
(cluster-ising) is straightforward, and the writing ofthe code takes
no more than a few hours.Itis the understanding, especially the opera-
tional handling ofprobabilities, which is difficulttoobtain.Itis onthis
point that wehave been focusing.
Inthiscontext, it is essential to realize that powerful Monte Carlo
methods which allow one to reach huge system sizes, and obtain mil-
lions of essentially independent samples, are the exception rather than
the rule.As stressed throughoutthis book, one often has to face severe
restrictions onthe number of statistically independent samples which
can be produced even during long runs.Moreover, even in cases where
Monte Carlo methods work well (as in the Ising model),there is tight
competition with other methods, such as transfer matrix approaches, ex-
act enumerations, and high-temperature expansions (which are usually,
however, less versatile). These methods only work forsmall lattices, but
they make upmuch ground with respect to the Monte Carlo approach
because they produce numerically exact results forsmall systems, and
can be extrapolated much better because they haveno statisticaluncer-
tainties.It takes dedication and programming skills, good understand-
ing, and fair judgment to find one’s way through this maze ofmodels
and methods.