3,546 research outputs found
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
Stiffness Exponents for Lattice Spin Glasses in Dimensions d=3,...,6
The stiffness exponents in the glass phase for lattice spin glasses in
dimensions are determined. To this end, we consider bond-diluted
lattices near the T=0 glass transition point . This transition for
discrete bond distributions occurs just above the bond percolation point
in each dimension. Numerics suggests that both points, and , seem to
share the same -expansion, at least for several leading orders, each
starting with . Hence, these lattice graphs have average connectivities
of near and exact graph-reduction methods become
very effective in eliminating recursively all spins of connectivity ,
allowing the treatment of lattices of lengths up to L=30 and with up to
spins. Using finite-size scaling, data for the defect energy width
over a range of in each dimension can be combined to
reach scaling regimes of about one decade in the scaling variable
. Accordingly, unprecedented accuracy is obtained for the
stiffness exponents compared to undiluted lattices (), where scaling is
far more limited. Surprisingly, scaling corrections typically are more benign
for diluted lattices. We find in for the stiffness exponents
, , and . The result for the
upper critical dimension, , suggest a mean-field value of .Comment: 8 pages, RevTex, 15 ps-figures included (see
http://www.physics.emory.edu/faculty/boettcher for related information
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
Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks
The Ising model with ferromagnetic couplings on the Hanoi networks is
analyzed with an exact renormalization group. In particular, the fixed-points
are determined and the renormalization-group flow for certain initial
conditions is analyzed. Hanoi networks combine a one-dimensional lattice
structure with a hierarchy of small-world bonds to create a mix of geometric
and mean-field properties. Generically, the small-world bonds result in
non-universal behavior, i.e. fixed points and scaling exponents that depend on
temperature and the initial choice of coupling strengths. It is shown that a
diversity of different behaviors can be observed with seemingly small changes
in the structure of the networks. Defining interpolating families of such
networks, we find tunable transitions between regimes with power-law and
certain essential singularities in the critical scaling of the correlation
length, similar to the so-called inverted Berezinskii-Kosterlitz-Thouless
transition previously observed only in scale-free or dense networks.Comment: 13 pages, revtex, 12 fig. incl.; fixed confusing labels, published
version. For related publications, see
http://www.physics.emory.edu/faculty/boettcher
Extremal Optimization at the Phase Transition of the 3-Coloring Problem
We investigate the phase transition of the 3-coloring problem on random
graphs, using the extremal optimization heuristic. 3-coloring is among the
hardest combinatorial optimization problems and is closely related to a 3-state
anti-ferromagnetic Potts model. Like many other such optimization problems, it
has been shown to exhibit a phase transition in its ground state behavior under
variation of a system parameter: the graph's mean vertex degree. This phase
transition is often associated with the instances of highest complexity. We use
extremal optimization to measure the ground state cost and the ``backbone'', an
order parameter related to ground state overlap, averaged over a large number
of instances near the transition for random graphs of size up to 512. For
graphs up to this size, benchmarks show that extremal optimization reaches
ground states and explores a sufficient number of them to give the correct
backbone value after about update steps. Finite size scaling gives
a critical mean degree value . Furthermore, the
exploration of the degenerate ground states indicates that the backbone order
parameter, measuring the constrainedness of the problem, exhibits a first-order
phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available
at http://www.physics.emory.edu/faculty/boettcher
Numerical Results for Ground States of Mean-Field Spin Glasses at low Connectivities
An extensive list of results for the ground state properties of spin glasses
on random graphs is presented. These results provide a timely benchmark for
currently developing theoretical techniques based on replica symmetry breaking
that are being tested on mean-field models at low connectivity. Comparison with
existing replica results for such models verifies the strength of those
techniques. Yet, we find that spin glasses on fixed-connectivity graphs (Bethe
lattices) exhibit a richer phenomenology than has been anticipated by theory.
Our data prove to be sufficiently accurate to speculate about some exact
results.Comment: 4 pages, RevTex4, 5 ps-figures included, related papers available at
http://www.physics.emory.edu/faculty/boettcher
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