511 research outputs found
Critical Droplets and Phase Transitions in Two Dimensions
In two space dimensions, the percolation point of the pure-site clusters of
the Ising model coincides with the critical point T_c of the thermal transition
and the percolation exponents belong to a special universality class. By
introducing a bond probability p_B<1, the corresponding site-bond clusters keep
on percolating at T_c and the exponents do not change, until
p_B=p_CK=1-exp(-2J/kT): for this special expression of the bond weight the
critical percolation exponents switch to the 2D Ising universality class. We
show here that the result is valid for a wide class of bidimensional models
with a continuous magnetization transition: there is a critical bond
probability p_c such that, for any p_B>=p_c, the onset of percolation of the
site-bond clusters coincides with the critical point of the thermal transition.
The percolation exponents are the same for p_c<p_B<=1 but, for p_B=p_c, they
suddenly change to the thermal exponents, so that the corresponding clusters
are critical droplets of the phase transition. Our result is based on Monte
Carlo simulations of various systems near criticality.Comment: Final version for publication, minor changes, figures adde
SLE in the three-state Potts model - a numerical study
The scaling limit of the spin cluster boundaries of the Ising model with
domain wall boundary conditions is SLE with kappa=3. We hypothesise that the
three-state Potts model with appropriate boundary conditions has spin cluster
boundaries which are also SLE in the scaling limit, but with kappa=10/3. To
test this, we generate samples using the Wolff algorithm and test them against
predictions of SLE: we examine the statistics of the Loewner driving function,
estimate the fractal dimension and test against Schramm's formula. The results
are in support of our hypothesis.Comment: 32 pages, 41 figure
Exact sampling from non-attractive distributions using summary states
Propp and Wilson's method of coupling from the past allows one to efficiently
generate exact samples from attractive statistical distributions (e.g., the
ferromagnetic Ising model). This method may be generalized to non-attractive
distributions by the use of summary states, as first described by Huber. Using
this method, we present exact samples from a frustrated antiferromagnetic
triangular Ising model and the antiferromagnetic q=3 Potts model. We discuss
the advantages and limitations of the method of summary states for practical
sampling, paying particular attention to the slowing down of the algorithm at
low temperature. In particular, we show that such a slowing down can occur in
the absence of a physical phase transition.Comment: 5 pages, 6 EPS figures, REVTeX; additional information at
http://wol.ra.phy.cam.ac.uk/mackay/exac
Extended Defects in the Potts-Percolation Model of a Solid: Renormalization Group and Monte Carlo Analysis
We extend the model of a 2 solid to include a line of defects. Neighboring
atoms on the defect line are connected by ?springs? of different strength and
different cohesive energy with respect to the rest of the system. Using the
Migdal-Kadanoff renormalization group we show that the elastic energy is an
irrelevant field at the bulk critical point. For zero elastic energy this model
reduces to the Potts model. By using Monte Carlo simulations of the 3- and
4-state Potts model on a square lattice with a line of defects, we confirm the
renormalization-group prediction that for a defect interaction larger than the
bulk interaction the order parameter of the defect line changes discontinuously
while the defect energy varies continuously as a function of temperature at the
bulk critical temperature.Comment: 13 figures, 17 page
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
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
Potts-Percolation-Gauss Model of a Solid
We study a statistical mechanics model of a solid. Neighboring atoms are
connected by Hookian springs. If the energy is larger than a threshold the
"spring" is more likely to fail, while if the energy is lower than the
threshold the spring is more likely to be alive. The phase diagram and
thermodynamic quantities, such as free energy, numbers of bonds and clusters,
and their fluctuations, are determined using renormalization-group and
Monte-Carlo techniques.Comment: 10 pages, 12 figure
Critical slowing down in polynomial time algorithms
Combinatorial optimization algorithms which compute exact ground state
configurations in disordered magnets are seen to exhibit critical slowing down
at zero temperature phase transitions. Using arguments based on the physical
picture of the model, including vanishing stiffness on scales beyond the
correlation length and the ground state degeneracy, the number of operations
carried out by one such algorithm, the push-relabel algorithm for the random
field Ising model, can be estimated. Some scaling can also be predicted for the
2D spin glass.Comment: 4 pp., 3 fig
Microcanonical cluster algorithms
I propose a numerical simulation algorithm for statistical systems which
combines a microcanonical transfer of energy with global changes in clusters of
spins. The advantages of the cluster approach near a critical point augment the
speed increases associated with multi-spin coding in the microcanonical
approach. The method also provides a limited ability to tune the average
cluster size.Comment: 10 page
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