576 research outputs found
Coloring random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters and where the
proliferation of metastable states is responsible for the onset of complexity
in local search algorithms.Comment: 4 pages, 1 figure, version to app. in PR
Polynomial iterative algorithms for coloring and analyzing random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters. Furthermore, we
extended our considerations to the case of single instances showing consistent
results. This lead us to propose a new algorithm able to color in polynomial
time random graphs in the hard but colorable region, i.e when .Comment: 23 pages, 10 eps figure
Computing a Knot Invariant as a Constraint Satisfaction Problem
We point out the connection between mathematical knot theory and spin
glass/search problem. In particular, we present a statistical mechanical
formulation of the problem of computing a knot invariant; p-colorability
problem, which provides an algorithm to find the solution. The method also
allows one to get some deeper insight into the structural complexity of knots,
which is expected to be related with the landscape structure of constraint
satisfaction problem.Comment: 6 pages, 3 figures, submitted to short note in Journal of Physical
Society of Japa
Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model
We present and solve the Replica Symmetric equations in the context of the
Replica Cluster Variational Method for the 2D random bond Ising model
(including the 2D Edwards-Anderson spin glass model). First we solve a
linearized version of these equations to obtain the phase diagrams of the model
on the square and triangular lattices. In both cases the spin-glass transition
temperatures and the tricritical point estimations improve largely over the
Bethe predictions. Moreover, we show that this phase diagram is consistent with
the behavior of inference algorithms on single instances of the problem.
Finally, we present a method to consistently find approximate solutions to the
equations in the glassy phase. The method is applied to the triangular lattice
down to T=0, also in the presence of an external field.Comment: 22 pages, 11 figure
Dynamics of systems with isotropic competing interactions in an external field: a Langevin approach
We study the Langevin dynamics of a ferromagnetic Ginzburg-Landau Hamiltonian
with a competing long-range repulsive term in the presence of an external
magnetic field. The model is analytically solved within the self consistent
Hartree approximation for two different initial conditions: disordered or zero
field cooled (ZFC), and fully magnetized or field cooled (FC). To test the
predictions of the approximation we develop a suitable numerical scheme to
ensure the isotropic nature of the interactions. Both the analytical approach
and the numerical simulations of two-dimensional finite systems confirm a
simple aging scenario at zero temperature and zero field. At zero temperature a
critical field is found below which the initial conditions are relevant
for the long time dynamics of the system. For a logarithmic growth of
modulated domains is found in the numerical simulations but this behavior is
not captured by the analytical approach which predicts a growth law at
Characterizing and Improving Generalized Belief Propagation Algorithms on the 2D Edwards-Anderson Model
We study the performance of different message passing algorithms in the two
dimensional Edwards Anderson model. We show that the standard Belief
Propagation (BP) algorithm converges only at high temperature to a paramagnetic
solution. Then, we test a Generalized Belief Propagation (GBP) algorithm,
derived from a Cluster Variational Method (CVM) at the plaquette level. We
compare its performance with BP and with other algorithms derived under the
same approximation: Double Loop (DL) and a two-ways message passing algorithm
(HAK). The plaquette-CVM approximation improves BP in at least three ways: the
quality of the paramagnetic solution at high temperatures, a better estimate
(lower) for the critical temperature, and the fact that the GBP message passing
algorithm converges also to non paramagnetic solutions. The lack of convergence
of the standard GBP message passing algorithm at low temperatures seems to be
related to the implementation details and not to the appearance of long range
order. In fact, we prove that a gauge invariance of the constrained CVM free
energy can be exploited to derive a new message passing algorithm which
converges at even lower temperatures. In all its region of convergence this new
algorithm is faster than HAK and DL by some orders of magnitude.Comment: 19 pages, 13 figure
Magnetic field chaos in the SK Model
We study the Sherrington--Kirkpatrick model, both above and below the De
Almeida Thouless line, by using a modified version of the Parallel Tempering
algorithm in which the system is allowed to move between different values of
the magnetic field h. The behavior of the probability distribution of the
overlap between two replicas at different values of the magnetic field h_0 and
h_1 gives clear evidence for the presence of magnetic field chaos already for
moderate system sizes, in contrast to the case of temperature chaos, which is
not visible on system sizes that can currently be thermalized.Comment: Latex, 16 pages including 20 postscript figure
Coupled surface polaritons and the Casimir force
The Casimir force between metallic plates made of realistic materials is
evaluated for distances in the nanometer range. A spectrum over real
frequencies is introduced and shows narrow peaks due to surface resonances
(plasmon polaritons or phonon polaritons) that are coupled across the vacuum
gap. We demonstrate that the Casimir force originates from the attraction
(repulsion) due to the corresponding symmetric (antisymmetric) eigenmodes,
respectively. This picture is used to derive a simple analytical estimate of
the Casimir force at short distances. We recover the result known for Drude
metals without absorption and compute the correction for weakly absorbing
materials.Comment: revised version submitted to Phys. Rev. A, 06 November 200
Genetic diversity of the root-knot nematode Meloidogyne enterolobii and development of a SCAR marker for this guava-damaging species.
Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms
Stochastic local search algorithms are frequently used to numerically solve
hard combinatorial optimization or decision problems. We give numerical and
approximate analytical descriptions of the dynamics of such algorithms applied
to random satisfiability problems. We find two different dynamical regimes,
depending on the number of constraints per variable: For low constraintness,
the problems are solved efficiently, i.e. in linear time. For higher
constraintness, the solution times become exponential. We observe that the
dynamical behavior is characterized by a fast equilibration and fluctuations
around this equilibrium. If the algorithm runs long enough, an exponentially
rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003
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