400 research outputs found
Efficient Monte Carlo algorithm and high-precision results for percolation
We present a new Monte Carlo algorithm for studying site or bond percolation
on any lattice. The algorithm allows us to calculate quantities such as the
cluster size distribution or spanning probability over the entire range of site
or bond occupation probabilities from zero to one in a single run which takes
an amount of time scaling linearly with the number of sites on the lattice. We
use our algorithm to determine that the percolation transition occurs at
occupation probability 0.59274621(13) for site percolation on the square
lattice and to provide clear numerical confirmation of the conjectured
4/3-power stretched-exponential tails in the spanning probability functions.Comment: 8 pages, including 3 postscript figures, minor corrections in this
version, plus updated figures for the position of the percolation transitio
Precise determination of the bond percolation thresholds and finite-size scaling corrections for the s.c., f.c.c., and b.c.c. lattices
Extensive Monte-Carlo simulations were performed to study bond percolation on
the simple cubic (s.c.), face-centered cubic (f.c.c.), and body-centered cubic
(b.c.c.) lattices, using an epidemic kind of approach. These simulations
provide very precise values of the critical thresholds for each of the
lattices: pc(s.c.) = 0.248 812 6(5), pc(f.c.c.) = 0.120 163 5(10), and
pc(b.c.c.) = 0.180 287 5(10). For p close to pc, the results follow the
expected finite-size and scaling behavior, with values for the Fisher exponent
(2.189(2)), the finite-size correction exponent (0.64(2)), and
the scaling function exponent (0.445(1)) confirmed to be universal.Comment: 16 pgs, 7 figures, LaTeX, to be published in Phys. Rev.
Equivalence of stationary state ensembles
We show that the contact process in an ensemble with conserved total particle
number, as simulated recently by Tome and de Oliveira [Phys. Rev. Lett. 86
(2001) 5463], is equivalent to the ordinary contact process, in agreement with
what the authors assumed and believed. Similar conserved ensembles and
equivalence proofs are easily constructed for other models.Comment: 5 pages, no figure
The Largest Cluster in Subcritical Percolation
The statistical behavior of the size (or mass) of the largest cluster in
subcritical percolation on a finite lattice of size is investigated (below
the upper critical dimension, presumably ). It is argued that as the cumulative distribution function converges to the Fisher-Tippett
(or Gumbel) distribution in a certain weak sense (when suitably
normalized). The mean grows like , where is a
``crossover size''. The standard deviation is bounded near with persistent fluctuations due to discreteness. These
predictions are verified by Monte Carlo simulations on square lattices of
up to 30 million sites, which also reveal finite-size scaling. The results are
explained in terms of a flow in the space of probability distributions as . The subcritical segment of the physical manifold ()
approaches a line of limit cycles where the flow is approximately described by
a ``renormalization group'' from the classical theory of extreme order
statistics.Comment: 16 pages, 5 figs, expanded version to appear in Phys Rev
Geometric Random Inner Products: A New Family of Tests for Random Number Generators
We present a new computational scheme, GRIP (Geometric Random Inner
Products), for testing the quality of random number generators. The GRIP
formalism utilizes geometric probability techniques to calculate the average
scalar products of random vectors generated in geometric objects, such as
circles and spheres. We show that these average scalar products define a family
of geometric constants which can be used to evaluate the quality of random
number generators. We explicitly apply the GRIP tests to several random number
generators frequently used in Monte Carlo simulations, and demonstrate a new
statistical property for good random number generators
Random sequential adsorption on a dashed line
We study analytically and numerically a model of random sequential adsorption
(RSA) of segments on a line, subject to some constraints suggested by two kinds
of physical situations:
- deposition of dimers on a lattice where the sites have a spatial extension;
- deposition of extended particles which must overlap one (or several)
adsorbing sites on the substrate.
Both systems involve discrete and continuous degrees of freedom, and, in one
dimension, are equivalent to our model, which depends on one length parameter.
When this parameter is varied, the model interpolates between a variety of
known situations : monomers on a lattice, "car-parking" problem, dimers on a
lattice. An analysis of the long-time behaviour of the coverage as a function
of the parameter exhibits an anomalous 1/t^2 approach to the jamming limit at
the transition point between the fast exponential kinetics, characteristic of
the lattice model, and the 1/t law of the continuous one.Comment: 14 pages (Latex) + 4 Postscript figure
Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation
We study nonequilibrium phase transitions in a mass-aggregation model which
allows for diffusion, aggregation on contact, dissociation, adsorption and
desorption of unit masses. We analyse two limits explicitly. In the first case
mass is locally conserved whereas in the second case local conservation is
violated. In both cases the system undergoes a dynamical phase transition in
all dimensions. In the first case, the steady state mass distribution decays
exponentially for large mass in one phase, and develops an infinite aggregate
in addition to a power-law mass decay in the other phase. In the second case,
the transition is similar except that the infinite aggregate is missing.Comment: Major revision of tex
Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation
Extensive Monte-Carlo simulations were performed to evaluate the excess
number of clusters and the crossing probability function for three-dimensional
percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and
body-centered cubic (b.c.c.) lattices. Systems L x L x L' with L' >> L were
studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The
excess number of clusters per unit length was confirmed to be a
universal quantity with a value . Likewise, the
critical crossing probability in the L' direction, with periodic boundary
conditions in the L x L plane, was found to follow a universal exponential
decay as a function of r = L'/L for large r. Simulations were also carried out
to find new precise values of the critical thresholds for site percolation on
the f.c.c. and b.c.c. lattices, yielding , .Comment: 14 pages, 7 figures, LaTeX, submitted to J. Phys. A: Math. Gen, added
references, corrected typo
Site percolation and random walks on d-dimensional Kagome lattices
The site percolation problem is studied on d-dimensional generalisations of
the Kagome' lattice. These lattices are isotropic and have the same
coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d.
The site percolation thresholds are calculated numerically for d= 3, 4, 5, and
6. The scaling of these thresholds as a function of dimension d, or
alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead
of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually
assumed to hold for all lattices in high dimensions. A series expansion is
calculated, in order to understand the different behaviour of the Kagome'
lattice. The return probability of a random walker on these lattices is also
shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices
these results imply p_c ~ 1/(q-1).Comment: 11 pages, LaTeX, 8 figures (EPS format), submitted to J. Phys.
Domain Growth in a 1-D Driven Diffusive System
The low-temperature coarsening dynamics of a one-dimensional Ising model,
with conserved magnetisation and subject to a small external driving force, is
studied analytically in the limit where the volume fraction \mu of the minority
phase is small, and numerically for general \mu. The mean domain size L(t)
grows as t^{1/2} in all cases, and the domain-size distribution for domains of
one sign is very well described by the form P_l(l) \propto
(l/L^3)\exp[-\lambda(\mu)(l^2/L^2)], which is exact for small \mu (and possibly
for all \mu). The persistence exponent for the minority phase has the value 3/2
for \mu \to 0.Comment: 8 pages, REVTeX, 7 Postscript figures, uses multicol.sty and
epsf.sty. Submitted to Phys. Rev.
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