2,454 research outputs found
The backtracking survey propagation algorithm for solving random K-SAT problems
Discrete combinatorial optimization has a central role in many scientific
disciplines, however, for hard problems we lack linear time algorithms that
would allow us to solve very large instances. Moreover, it is still unclear
what are the key features that make a discrete combinatorial optimization
problem hard to solve. Here we study random K-satisfiability problems with
, which are known to be very hard close to the SAT-UNSAT threshold,
where problems stop having solutions. We show that the backtracking survey
propagation algorithm, in a time practically linear in the problem size, is
able to find solutions very close to the threshold, in a region unreachable by
any other algorithm. All solutions found have no frozen variables, thus
supporting the conjecture that only unfrozen solutions can be found in linear
time, and that a problem becomes impossible to solve in linear time when all
solutions contain frozen variables.Comment: 11 pages, 10 figures. v2: data largely improved and manuscript
rewritte
The crossover region between long-range and short-range interactions for the critical exponents
It is well know that systems with an interaction decaying as a power of the
distance may have critical exponents that are different from those of
short-range systems. The boundary between long-range and short-range is known,
however the behavior in the crossover region is not well understood. In this
paper we propose a general form for the crossover function and we compute it in
a particular limit. We compare our predictions with the results of numerical
simulations for two-dimensional long-range percolation.Comment: 17 pages, 6 figure
The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism
The ferromagnetic XY model on sparse random graphs in a randomly oriented
field is analyzed via the belief propagation algorithm. At variance with the
fully connected case and with the random field Ising model on the same
topology, we find strong evidences of a tiny region with Replica Symmetry
Breaking (RSB) in the limit of very low temperatures. This RSB phase is robust
against different choices of the external field direction, while it rapidly
vanishes when increasing the graph mean degree, the temperature or the
directional bias in the external field. The crucial ingredients to have such a
RSB phase seem to be the continuous nature of vector spins, mostly preserved by
the O(2)-invariant random field, and the strong spatial heterogeneity, due to
graph sparsity. We also uncover that the ferromagnetic phase can be marginally
stable despite the presence of the random field. Finally, we study the proper
correlation functions approaching the critical points to identify the ones that
become more critical.Comment: 14 pages, 9 figure
One-loop topological expansion for spin glasses in the large connectivity limit
We apply for the first time a new one-loop topological expansion around the
Bethe solution to the spin-glass model with field in the high connectivity
limit, following the methodological scheme proposed in a recent work. The
results are completely equivalent to the well known ones, found by standard
field theoretical expansion around the fully connected model (Bray and Roberts
1980, and following works). However this method has the advantage that the
starting point is the original Hamiltonian of the model, with no need to define
an associated field theory, nor to know the initial values of the couplings,
and the computations have a clear and simple physical meaning. Moreover this
new method can also be applied in the case of zero temperature, when the Bethe
model has a transition in field, contrary to the fully connected model that is
always in the spin glass phase. Sharing with finite dimensional model the
finite connectivity properties, the Bethe lattice is clearly a better starting
point for an expansion with respect to the fully connected model. The present
work is a first step towards the generalization of this new expansion to more
difficult and interesting cases as the zero-temperature limit, where the
expansion could lead to different results with respect to the standard one.Comment: 8 pages, 1 figur
Ensemble renormalization group for disordered systems
We propose and study a renormalization group transformation that can be used
also for models with strong quenched disorder, like spin glasses. The method is
based on a mapping between disorder distributions, chosen such as to keep some
physical properties (e.g., the ratio of correlations averaged over the
ensemble) invariant under the transformation. We validate this ensemble
renormalization group by applying it to the hierarchical model (both the
diluted ferromagnetic version and the spin glass version), finding results in
agreement with Monte Carlo simulations.Comment: 7 pages, 10 figure
Generalized off-equilibrium fluctuation-dissipation relations in random Ising systems
We show that the numerical method based on the off-equilibrium
fluctuation-dissipation relation does work and is very useful and powerful in
the study of disordered systems which show a very slow dynamics. We have
verified that it gives the right information in the known cases (diluted
ferromagnets and random field Ising model far from the critical point) and we
used it to obtain more convincing results on the frozen phase of
finite-dimensional spin glasses. Moreover we used it to study the Griffiths
phase of the diluted and the random field Ising models.Comment: 20 pages, 10 figures, uses epsfig.sty. Partially presented at
StatPhys XX in a talk by one of the authors (FRT). Added 1 reference in the
new versio
- …