397 research outputs found
Oriented Percolation in One-Dimensional 1/|x-y|^2 Percolation Models
We consider independent edge percolation models on Z, with edge occupation
probabilities p_ = p if |x-y| = 1, 1 - exp{- beta / |x-y|^2} otherwise. We
prove that oriented percolation occurs when beta > 1 provided p is chosen
sufficiently close to 1, answering a question posed in [Commun. Math. Phys.
104, 547 (1986)]. The proof is based on multi-scale analysis.Comment: 19 pages, 2 figures. See also Commentary on J. Stat. Phys. 150,
804-805 (2013), DOI 10.1007/s10955-013-0702-
Local and cluster critical dynamics of the 3d random-site Ising model
We present the results of Monte Carlo simulations for the critical dynamics
of the three-dimensional site-diluted quenched Ising model. Three different
dynamics are considered, these correspond to the local update Metropolis scheme
as well as to the Swendsen-Wang and Wolff cluster algorithms. The lattice sizes
of L=10-96 are analysed by a finite-size-scaling technique. The site dilution
concentration p=0.85 was chosen to minimize the correction-to-scaling effects.
We calculate numerical values of the dynamical critical exponents for the
integrated and exponential autocorrelation times for energy and magnetization.
As expected, cluster algorithms are characterized by lower values of dynamical
critical exponent than the local one: also in the case of dilution critical
slowing down is more pronounced for the Metropolis algorithm. However, the
striking feature of our estimates is that they suggest that dilution leads to
decrease of the dynamical critical exponent for the cluster algorithms. This
phenomenon is quite opposite to the local dynamics, where dilution enhances
critical slowing down.Comment: 24 pages, 16 figures, style file include
Predictions of bond percolation thresholds for the kagom\'e and Archimedean lattices
Here we show how the recent exact determination of the bond percolation
threshold for the martini lattice can be used to provide approximations to the
unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one
of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2)
bond problems, and the other gives estimates of for the homogeneous
kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively
agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure
A necklace of Wulff shapes
In a probabilistic model of a film over a disordered substrate, Monte-Carlo
simulations show that the film hangs from peaks of the substrate. The film
profile is well approximated by a necklace of Wulff shapes. Such a necklace can
be obtained as the infimum of a collection of Wulff shapes resting on the
substrate. When the random substrate is given by iid heights with exponential
distribution, we prove estimates on the probability density of the resulting
peaks, at small density
Dynamic critical behavior of the Swendsen--Wang Algorithm for the three-dimensional Ising model
We have performed a high-precision Monte Carlo study of the dynamic critical
behavior of the Swendsen-Wang algorithm for the three-dimensional Ising model
at the critical point. For the dynamic critical exponents associated to the
integrated autocorrelation times of the "energy-like" observables, we find
z_{int,N} = z_{int,E} = z_{int,E'} = 0.459 +- 0.005 +- 0.025, where the first
error bar represents statistical error (68% confidence interval) and the second
error bar represents possible systematic error due to corrections to scaling
(68% subjective confidence interval). For the "susceptibility-like"
observables, we find z_{int,M^2} = z_{int,S_2} = 0.443 +- 0.005 +- 0.030. For
the dynamic critical exponent associated to the exponential autocorrelation
time, we find z_{exp} \approx 0.481. Our data are consistent with the
Coddington-Baillie conjecture z_{SW} = \beta/\nu \approx 0.5183, especially if
it is interpreted as referring to z_{exp}.Comment: LaTex2e, 39 pages including 5 figure
Poisson approximations for the Ising model
A -dimensional Ising model on a lattice torus is considered. As the size
of the lattice tends to infinity, a Poisson approximation is given for the
distribution of the number of copies in the lattice of any given local
configuration, provided the magnetic field tends to and the
pair potential remains fixed. Using the Stein-Chen method, a bound is given
for the total variation error in the ferromagnetic case.Comment: 25 pages, 1 figur
Phase transitions with four-spin interactions
Using an extended Lee-Yang theorem and GKS correlation inequalities, we
prove, for a class of ferromagnetic multi-spin interactions, that they will
have a phase transition(and spontaneous magnetization) if, and only if, the
external field (and the temperature is low enough). We also show the
absence of phase transitions for some nonferromagnetic interactions. The FKG
inequalities are shown to hold for a larger class of multi-spin interactions
Rejection-free Geometric Cluster Algorithm for Complex Fluids
We present a novel, generally applicable Monte Carlo algorithm for the
simulation of fluid systems. Geometric transformations are used to identify
clusters of particles in such a manner that every cluster move is accepted,
irrespective of the nature of the pair interactions. The rejection-free and
non-local nature of the algorithm make it particularly suitable for the
efficient simulation of complex fluids with components of widely varying size,
such as colloidal mixtures. Compared to conventional simulation algorithms,
typical efficiency improvements amount to several orders of magnitude
Percolation on the average and spontaneous magnetization for q-states Potts model on graph
We prove that the q-states Potts model on graph is spontaneously magnetized
at finite temperature if and only if the graph presents percolation on the
average. Percolation on the average is a combinatorial problem defined by
averaging over all the sites of the graph the probability of belonging to a
cluster of a given size. In the paper we obtain an inequality between this
average probability and the average magnetization, which is a typical extensive
function describing the thermodynamic behaviour of the model
Conformal Invariance in Percolation, Self-Avoiding Walks and Related Problems
Over the years, problems like percolation and self-avoiding walks have
provided important testing grounds for our understanding of the nature of the
critical state. I describe some very recent ideas, as well as some older ones,
which cast light both on these problems themselves and on the quantum field
theories to which they correspond. These ideas come from conformal field
theory, Coulomb gas mappings, and stochastic Loewner evolution.Comment: Plenary talk given at TH-2002, Paris. 21 pages, 9 figure
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