105 research outputs found
Corrections to scaling for percolative conduction: anomalous behavior at small L
Recently Grassberger has shown that the correction to scaling for the
conductance of a bond percolation network on a square lattice is a nonmonotonic
function of the linear lattice dimension with a minimum at , while this
anomalous behavior is not present in the site percolation networks. We perform
a high precision numerical study of the bond percolation random resistor
networks on the square, triangular and honeycomb lattices to further examine
this result. We use the arithmetic, geometric and harmonic means to obtain the
conductance and find that the qualitative behavior does not change: it is not
related to the shape of the conductance distribution for small system sizes. We
show that the anomaly at small L is absent on the triangular and honeycomb
networks. We suggest that the nonmonotonic behavior is an artifact of
approximating the continuous system for which the theory is formulated by a
discrete one which can be simulated on a computer. We show that by slightly
changing the definition of the linear lattice size we can eliminate the minimum
at small L without significantly affecting the large L limit.Comment: 3 pages, 4 figures;slightly expanded, 2 figures added. Accepted for
publishing in Phys. Rev.
Percolation on two- and three-dimensional lattices
In this work we apply a highly efficient Monte Carlo algorithm recently
proposed by Newman and Ziff to treat percolation problems. The site and bond
percolation are studied on a number of lattices in two and three dimensions.
Quite good results for the wrapping probabilities, correlation length critical
exponent and critical concentration are obtained for the square, simple cubic,
HCP and hexagonal lattices by using relatively small systems. We also confirm
the universal aspect of the wrapping probabilities regarding site and bond
dilution.Comment: 15 pages, 6 figures, 3 table
Continuous Percolation Phase Transitions of Two-dimensional Lattice Networks under a Generalized Achlioptas Process
The percolation phase transitions of two-dimensional lattice networks under a
generalized Achlioptas process (GAP) are investigated. During the GAP, two
edges are chosen randomly from the lattice and the edge with minimum product of
the two connecting cluster sizes is taken as the next occupied bond with a
probability . At , the GAP becomes the random growth model and leads
to the minority product rule at . Using the finite-size scaling analysis,
we find that the percolation phase transitions of these systems with are always continuous and their critical exponents depend on .
Therefore, the universality class of the critical phenomena in two-dimensional
lattice networks under the GAP is related to the probability parameter in
addition.Comment: 7 pages, 14 figures, accepted for publication in Eur. Phys. J.
Scaling of loop-erased walks in 2 to 4 dimensions
We simulate loop-erased random walks on simple (hyper-)cubic lattices of
dimensions 2,3, and 4. These simulations were mainly motivated to test recent
two loop renormalization group predictions for logarithmic corrections in
, simulations in lower dimensions were done for completeness and in order
to test the algorithm. In , we verify with high precision the prediction
, where the number of steps after erasure scales with the number
of steps before erasure as . In we again find a power law,
but with an exponent different from the one found in the most precise previous
simulations: . Finally, we see clear deviations from the
naive scaling in . While they agree only qualitatively with the
leading logarithmic corrections predicted by several authors, their agreement
with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure
Recent advances and open challenges in percolation
Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation
The critical amplitude ratio of the susceptibility in the random-site two-dimensional Ising model
We present a new way of probing the universality class of the site-diluted
two-dimensional Ising model. We analyse Monte Carlo data for the magnetic
susceptibility, introducing a new fitting procedure in the critical region
applicable even for a single sample with quenched disorder. This gives us the
possibility to fit simultaneously the critical exponent, the critical amplitude
and the sample dependent pseudo-critical temperature. The critical amplitude
ratio of the magnetic susceptibility is seen to be independent of the
concentration of the empty sites for all investigated values of . At the same time the average effective exponent is found
to vary with the concentration , which may be argued to be due to
logarithmic corrections to the power law of the pure system. This corrections
are canceled in the susceptibility amplitude ratio as predicted by theory. The
central charge of the corresponding field theory was computed and compared well
with the theoretical predictions.Comment: 6 pages, 4 figure
Universal features of the order-parameter fluctuations : reversible and irreversible aggregation
We discuss the universal scaling laws of order parameter fluctuations in any
system in which the second-order critical behaviour can be identified. These
scaling laws can be derived rigorously for equilibrium systems when combined
with the finite-size scaling analysis. The relation between order parameter,
criticality and scaling law of fluctuations has been established and the
connexion between the scaling function and the critical exponents has been
found. We give examples in out-of-equilibrium aggregation models such as the
Smoluchowski kinetic equations, or of at-equilibrium Ising and percolation
models.Comment: 19 pages, 10 figure
Universality of the Crossing Probability for the Potts Model for q=1,2,3,4
The universality of the crossing probability of a system to
percolate only in the horizontal direction, was investigated numerically by
using a cluster Monte-Carlo algorithm for the -state Potts model for
and for percolation . We check the percolation through
Fortuin-Kasteleyn clusters near the critical point on the square lattice by
using representation of the Potts model as the correlated site-bond percolation
model. It was shown that probability of a system to percolate only in the
horizontal direction has universal form for
as a function of the scaling variable . Here,
is the probability of a bond to be closed, is the
nonuniversal crossing amplitude, is the nonuniversal metric factor,
is the nonuniversal scaling index, is the correlation
length index.
The universal function . Nonuniversal scaling factors
were found numerically.Comment: 15 pages, 3 figures, revtex4b, (minor errors in text fixed,
journal-ref added
Critical Percolation in High Dimensions
We present Monte Carlo estimates for site and bond percolation thresholds in
simple hypercubic lattices with 4 to 13 dimensions. For d<6 they are
preliminary, for d >= 6 they are between 20 to 10^4 times more precise than the
best previous estimates. This was achieved by three ingredients: (i) simple and
fast hashing which allowed us to simulate clusters of millions of sites on
computers with less than 500 MB memory; (ii) a histogram method which allowed
us to obtain information for several p values from a single simulation; and
(iii) a new variance reduction technique which is especially efficient at high
dimensions where it reduces error bars by a factor up to approximately 30 and
more. Based on these data we propose a new scaling law for finite cluster size
corrections.Comment: 5 pages including figures, RevTe
A precise approximation for directed percolation in d=1+1
We introduce an approximation specific to a continuous model for directed
percolation, which is strictly equivalent to 1+1 dimensional directed bond
percolation. We find that the critical exponent associated to the order
parameter (percolation probability) is beta=(1-1/\sqrt{5})/2=0.276393202..., in
remarkable agreement with the best current numerical estimate beta=0.276486(8).Comment: 4 pages, 3 EPS figures; Submitted to Physical Review Letters v2:
minor typos + 1 major typo in Eq. (30) correcte
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