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, Mexico D.F.
(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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