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

that isthe second quantization of the operator M T on the energy Hilbert space (see [20,
Section 8] for details). It follows from (5.55) that
and hence that
k;(M) n j n 1 ? kL 2 ( ) = kM n k(UA) (5.56)
(;(M) j n 1 ? ) = (M) : (5.57)
Moreover, P is self-adjoint onL 2 ( ) [i.e. satis es detailed balance] if and onlyifM is
self-adjoint with respect to the energy inner product, i.e.
and inthis case
MA = AM T (5.58)
(;(M) j n 1 ? ) =k;(M) j n 1 ? k L 2 ( ) = (M) =kMk(UA) : (5.59)
In summary, wehave shown that the dynamic behavior of any stochastic linear
iteration is completely determined by the behavior of the corresponding deterministic
linear iteration. In particular, the exponential autocorrelation time exp (slowest decay
rate ofanyautocorrelation function) is given by
exp(;1= exp) = (M) (5.60)
and this decay rate isachieved by atleastone observable which is linear in the eld
'. In other words, the (worst-case) convergence rate ofthe Monte Carlo algorithm is
precisely equal to the (worst-case) convergence rate ofthe corresponding deterministic
iteration.
In particular, for Gaussian MGMC, the convergence proofs for deterministic multigrid
[32, 40, 41] combined with the arguments ofthe present section prove rigorously
that critical slowing-down is completely eliminated (at least for a W-cycle). That is,
the autocorrelation time of the MGMCmethod is bounded as criticalityisapproached
(empirically 1 ; 2).
6 Swendsen-Wang Algorithms
Avery di erent type of collective-mode algorithm was proposed two years ago 22 by
Swendsen and Wang [27] for Potts spinmodels. Since then, there has been an explosion
of work tryingtounderstand whythis algorithm works so well (andwhy it does not work
22 Note Added 1996: Now nearly ten years ago! The great interest in algorithms of Swendsen-
Wang type (frequently called cluster algorithms) has not abated: see the references cited in the \Notes
Added" below, plus many others. See also [109, 110] for reviews that are slightly more up-to-date than
the present notes (1990 rather than 1989).
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