668 research outputs found
Survey Propagation as local equilibrium equations
It has been shown experimentally that a decimation algorithm based on Survey
Propagation (SP) equations allows to solve efficiently some combinatorial
problems over random graphs. We show that these equations can be derived as
sum-product equations for the computation of marginals in an extended space
where the variables are allowed to take an additional value -- -- when they
are not forced by the combinatorial constraints. An appropriate ``local
equilibrium condition'' cost/energy function is introduced and its entropy is
shown to coincide with the expected logarithm of the number of clusters of
solutions as computed by SP. These results may help to clarify the geometrical
notion of clusters assumed by SP for the random K-SAT or random graph coloring
(where it is conjectured to be exact) and helps to explain which kind of
clustering operation or approximation is enforced in general/small sized models
in which it is known to be inexact.Comment: 13 pages, 3 figure
Spanning Trees in Random Satisfiability Problems
Working with tree graphs is always easier than with loopy ones and spanning
trees are the closest tree-like structures to a given graph. We find a
correspondence between the solutions of random K-satisfiability problem and
those of spanning trees in the associated factor graph. We introduce a modified
survey propagation algorithm which returns null edges of the factor graph and
helps us to find satisfiable spanning trees. This allows us to study
organization of satisfiable spanning trees in the space spanned by spanning
trees.Comment: 12 pages, 5 figures, published versio
No-go theorem on spontaneous parity breaking revisited
An essential assumption in the Vafa and Witten's theorem on P and CT
realization in vector-like theories concerns the existence of a free energy
density in Euclidean space in the presence of any external hermitian symmetry
breaking source. We show how this requires the previous assumption that the
symmetry is realized in the vacuum. Even if Vafa and Witten's conjecture is
plausible, actually a theorem is still lacking.Comment: Talk presented at LATTICE99(Theoretical Developments),3 pages. Latex
using espcrc2.st
Strengths and Weaknesses of Parallel Tempering
Parallel tempering, also known as replica exchange Monte Carlo, is studied in
the context of two simple free energy landscapes. The first is a double well
potential defined by two macrostates separated by a barrier. The second is a
`golf course' potential defined by microstates having two possible energies
with exponentially more high energy states than low energy states. The
equilibration time for replica exchange is analyzed for both systems. For the
double well system, parallel tempering with a number of replicas that scales as
the square root of the barrier height yields exponential speedup of the
equilibration time. On the other hand, replica exchange yields only marginal
speed-up for the golf course system. For the double well system, the free
energy difference between the two wells has a large effect on the equilibration
time. Nearly degenerate wells equilibrate much more slowly than strongly
asymmetric wells. It is proposed that this difference in equilibration time may
lead to a bias in measuring overlaps in spin glasses. These examples illustrate
the strengths and weaknesses of replica exchange and may serve as a guide for
understanding and improving the method in various applications.Comment: 18 pages, 4 figures. v2: typos fixed and wording changes to improve
clarit
Replica Symmetry Breaking and the Renormalization Group Theory of the Weakly Disordered Ferromagnet
We study the critical properties of the weakly disordered -component
ferromagnet in terms of the renormalization group (RG) theory generalized to
take into account the replica symmetry breaking (RSB) effects coming from the
multiple local minima solutions of the mean-field equations. It is shown that
for the traditional RG flows at dimensions , which are
usually considered as describing the disorder-induced universal critical
behavior, are unstable with respect to the RSB potentials as found in spin
glasses. It is demonstrated that for a general type of the Parisi RSB
structures there exists no stable fixed points, and the RG flows lead to the
{\it strong coupling regime} at the finite scale , where
is the small parameter describing the disorder. The physical concequences
of the obtained RG solutions are discussed. In particular, we argue, that
discovered RSB strong coupling phenomena indicate on the onset of a new spin
glass type critical behaviour in the temperature interval near . Possible relevance of the considered RSB effects for
the Griffith phase is also discussed.Comment: 32 pages, Late
Non-Degenerate Ultrametric Diffusion
General non-degenerate p-adic operators of ultrametric diffusion are
introduced. Bases of eigenvectors for the introduced operators are constructed
and the corresponding eigenvalues are computed. Properties of the corresponding
dynamics (i.e. of the ultrametric diffusion) are investigated.Comment: 19 pages, 3 figure
Glassy dynamics, metastability limit and crystal growth in a lattice spin model
We introduce a lattice spin model where frustration is due to multibody
interactions rather than quenched disorder in the Hamiltonian. The system has a
crystalline ground state and below the melting temperature displays a dynamic
behaviour typical of fragile glasses. However, the supercooled phase loses
stability at an effective spinodal temperature, and thanks to this the Kauzmann
paradox is resolved. Below the spinodal the system enters an off-equilibrium
regime corresponding to fast crystal nucleation followed by slow activated
crystal growth. In this phase and in a time region which is longer the lower
the temperature we observe a violation of the fluctuation-dissipation theorem
analogous to structural glasses. Moreover, we show that in this system there is
no qualitative difference between a locally stable glassy configuration and a
highly disordered polycrystal
Solvable model of a polymer in random media with long ranged disorder correlations
We present an exactly solvable model of a Gaussian (flexible) polymer chain
in a quenched random medium. This is the case when the random medium obeys very
long range quadratic correlations. The model is solved in spatial
dimensions using the replica method, and practically all the physical
properties of the chain can be found. In particular the difference between the
behavior of a chain that is free to move and a chain with one end fixed is
elucidated. The interesting finding is that a chain that is free to move in a
quadratically correlated random potential behaves like a free chain with , where is the end to end distance and is the length of the
chain, whereas for a chain anchored at one end . The exact
results are found to agree with an alternative numerical solution in
dimensions. The crossover from long ranged to short ranged correlations of the
disorder is also explored.Comment: REVTeX, 28 pages, 12 figures in eps forma
Persistence and Memory in Patchwork Dynamics for Glassy Models
Slow dynamics in disordered materials prohibits direct simulation of their
rich nonequilibrium behavior at large scales. "Patchwork dynamics" is
introduced to mimic relaxation over a very broad range of time scales by
equilibrating or optimizing directly on successive length scales. This dynamics
is used to study coarsening and to replicate memory effects for spin glasses
and random ferromagnets. It is also used to find, with high confidence, exact
ground states in large or toroidal samples.Comment: 4 pages, 4 figures; reference correctio
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