Correlation Length, Isotropy, and Meta-stable States
Ricardo Garcia-Pelayo(a) and Peter F. Stadler(b,c)
(a) Instituto de Fisica, Universidad Nacional Autonoma de Mexico,
(b) Institut f. Theoretische Chemie, Univ. Wien, Austria
(c) The Santa Fe Institute, Santa Fe, New Mexico, U.S.A.
A landscape is rugged if it has many local optima,
if it gives rise to short adaptive walks, and if it exhibits
a rapidly decreasing pair-correlation function (and hence if
it has a short correlation length). The ``correlation length
conjecture'' allows to estimate the number of meta-stable states from
the correlation length, provided the landscape is ``typical''.
Isotropy, originally introduced as a geometrical condition on the
covariance matrix of a random field, can be re-interpreted as a maximum
entropy condition that lends a precise meaning to the notion of a
``typical'' landscape. The XY-Hamiltonian, which violates isotropy only
to a relatively small extent, is an ideal model for investigating the
influence of anisotropies. Numerical estimates for the number of local
optima and predictions obtained from the correlation length conjecture
indeed show deviations that increase with the extent of anisotropies
in the model.
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