8,354 research outputs found
Optimizing at the Ergodic Edge
Using a simple, annealed model, some of the key features of the recently
introduced extremal optimization heuristic are demonstrated. In particular, it
is shown that the dynamics of local search possesses a generic critical point
under the variation of its sole parameter, separating phases of too greedy
(non-ergodic, jammed) and too random (ergodic) exploration. Comparison of
various local search methods within this model suggests that the existence of
the critical point is essential for the optimal performance of the heuristic.Comment: RevTex4, 17 pages, 3 ps-figures incl., for related information, see
http://www.physics.emory.edu/faculty/boettcher/publications.htm
Extremal Optimization for Sherrington-Kirkpatrick Spin Glasses
Extremal Optimization (EO), a new local search heuristic, is used to
approximate ground states of the mean-field spin glass model introduced by
Sherrington and Kirkpatrick. The implementation extends the applicability of EO
to systems with highly connected variables. Approximate ground states of
sufficient accuracy and with statistical significance are obtained for systems
with more than N=1000 variables using bonds. The data reproduces the
well-known Parisi solution for the average ground state energy of the model to
about 0.01%, providing a high degree of confidence in the heuristic. The
results support to less than 1% accuracy rational values of for
the finite-size correction exponent, and of for the fluctuation
exponent of the ground state energies, neither one of which has been obtained
analytically yet. The probability density function for ground state energies is
highly skewed and identical within numerical error to the one found for
Gaussian bonds. But comparison with infinite-range models of finite
connectivity shows that the skewness is connectivity-dependent.Comment: Substantially revised, several new results, 5 pages, 6 eps figures
included, (see http://www.physics.emory.edu/faculty/boettcher/ for related
information
Numerical Results for Ground States of Spin Glasses on Bethe Lattices
The average ground state energy and entropy for +/- J spin glasses on Bethe
lattices of connectivities k+1=3...,26 at T=0 are approximated numerically. To
obtain sufficient accuracy for large system sizes (up to n=2048), the Extremal
Optimization heuristic is employed which provides high-quality results not only
for the ground state energies per spin e_{k+1} but also for their entropies
s_{k+1}. The results show considerable quantitative differences between
lattices of even and odd connectivities. The results for the ground state
energies compare very well with recent one-step replica symmetry breaking
calculations. These energies can be scaled for all even connectivities k+1 to
within a fraction of a percent onto a simple functional form, e_{k+1} = E_{SK}
sqrt(k+1) - {2E_{SK}+sqrt(2)} / sqrt(k+1), where E_{SK} = -0.7633 is the ground
state energy for the broken replica symmetry in the Sherrington-Kirkpatrick
model. But this form is in conflict with perturbative calculations at large
k+1, which do not distinguish between even and odd connectivities. We find
non-zero entropies s_{k+1} at small connectivities. While s_{k+1} seems to
vanish asymptotically with 1/(k+1) for even connectivities, it is
indistinguishable from zero already for odd k+1 >= 9.Comment: 11 pages, RevTex4, 28 ps-figures included, related papers available
at http://www.physics.emory.edu/faculty/boettcher
Aging Exponents in Self-Organized Criticality
In a recent Letter [Phys. Rev. Lett. 79, 889 (1997) and cond-mat/9702054] we
have demonstrated that the avalanches in the Bak-Sneppen model display aging
behavior similar to glassy systems. Numerical results for temporal correlations
show a broad distribution with two distinct regimes separated by a time scale
which is related to the age of the avalanche. This dynamical breaking of
time-translational invariance results in a new critical exponent, . Here we
present results for from extensive numerical simulations of self-organized
critical models in and 2. We find and
for the Bak-Sneppen model, and our results suggest
for the analytically tractable multi-trade model in both dimensions.Comment: 8 pages RevTex, 8 ps-figures included. Improved presentation, as to
appear in PR
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