3,094 research outputs found
Universality in Random Walk Models with Birth and Death
Models of random walks are considered in which walkers are born at one
location and die at all other locations with uniform death rate. Steady-state
distributions of random walkers exhibit dimensionally dependent critical
behavior as a function of the birth rate. Exact analytical results for a
hyperspherical lattice yield a second-order phase transition with a nontrivial
critical exponent for all positive dimensions . Numerical studies
of hypercubic and fractal lattices indicate that these exact results are
universal. Implications for the adsorption transition of polymers at curved
interfaces are discussed.Comment: 11 pages, revtex, 2 postscript figure
Low-Temperature Excitations of Dilute Lattice Spin Glasses
A new approach to exploring low-temperature excitations in finite-dimensional
lattice spin glasses is proposed. By focusing on bond-diluted lattices just
above the percolation threshold, large system sizes can be obtained which
lead to enhanced scaling regimes and more accurate exponents. Furthermore, this
method in principle remains practical for any dimension, yielding exponents
that so far have been elusive. This approach is demonstrated by determining the
stiffness exponent for dimensions , (the upper critical dimension),
and . Key is the application of an exact reduction algorithm, which
eliminates a large fraction of spins, so that the reduced lattices never exceed
variables for sizes as large as L=30 in , L=9 in , or L=8
in . Finite size scaling analysis gives for ,
significantly improving on previous work. The results for and ,
and , are entirely new and are compared with
mean-field predictions made for d>=6.Comment: 7 pages, LaTex, 7 ps-figures included, added result for stiffness in
d=7, as to appear in Europhysics Letters (see
http://www.physics.emory.edu/faculty/boettcher/ for related information
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
d_c=4 is the upper critical dimension for the Bak-Sneppen model
Numerical results are presented indicating d_c=4 as the upper critical
dimension for the Bak-Sneppen evolution model. This finding agrees with
previous theoretical arguments, but contradicts a recent Letter [Phys. Rev.
Lett. 80, 5746-5749 (1998)] that placed d_c as high as d=8. In particular, we
find that avalanches are compact for all dimensions d<=4, and are fractal for
d>4. Under those conditions, scaling arguments predict a d_c=4, where
hyperscaling relations hold for d<=4. Other properties of avalanches, studied
for 1<=d<=6, corroborate this result. To this end, an improved numerical
algorithm is presented that is based on the equivalent branching process.Comment: 4 pages, RevTex4, as to appear in Phys. Rev. Lett., related papers
available at http://userwww.service.emory.edu/~sboettc
Extremal Optimization for Graph Partitioning
Extremal optimization is a new general-purpose method for approximating
solutions to hard optimization problems. We study the method in detail by way
of the NP-hard graph partitioning problem. We discuss the scaling behavior of
extremal optimization, focusing on the convergence of the average run as a
function of runtime and system size. The method has a single free parameter,
which we determine numerically and justify using a simple argument. Our
numerical results demonstrate that on random graphs, extremal optimization
maintains consistent accuracy for increasing system sizes, with an
approximation error decreasing over runtime roughly as a power law t^(-0.4). On
geometrically structured graphs, the scaling of results from the average run
suggests that these are far from optimal, with large fluctuations between
individual trials. But when only the best runs are considered, results
consistent with theoretical arguments are recovered.Comment: 34 pages, RevTex4, 1 table and 20 ps-figures included, related papers
available at http://www.physics.emory.edu/faculty/boettcher
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