286 research outputs found
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
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
Dynamic Critical Behavior of the Chayes-Machta Algorithm for the Random-Cluster Model. I. Two Dimensions
We study, via Monte Carlo simulation, the dynamic critical behavior of the
Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which
generalizes the Swendsen-Wang dynamics for the q-state Potts ferromagnet to
non-integer q \ge 1. We consider spatial dimension d=2 and 1.25 \le q \le 4 in
steps of 0.25, on lattices up to 1024^2, and obtain estimates for the dynamic
critical exponent z_{CM}. We present evidence that when 1 \le q \lesssim 1.95
the Ossola-Sokal conjecture z_{CM} \ge \beta/\nu is violated, though we also
present plausible fits compatible with this conjecture. We show that the
Li-Sokal bound z_{CM} \ge \alpha/\nu is close to being sharp over the entire
range 1 \le q \le 4, but is probably non-sharp by a power. As a byproduct of
our work, we also obtain evidence concerning the corrections to scaling in
static observables.Comment: LaTeX2e, 75 pages including 26 Postscript figure
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
Mean Field Behavior of Cluster Dynamics
The dynamic behavior of cluster algorithms is analyzed in the classical mean
field limit. Rigorous analytical results below establish that the dynamic
exponent has the value for the Swendsen-Wang algorithm and
for the Wolff algorithm.
An efficient Monte Carlo implementation is introduced, adapted for using
these algorithms for fully connected graphs. Extensive simulations both above
and below demonstrate scaling and evaluate the finite-size scaling
function by means of a rather impressive collapse of the data.Comment: Revtex, 9 pages with 7 figure
Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
Denoting as the chromatic polynomial for coloring an -vertex
graph with colors, and considering the limiting function , a fundamental question in graph theory is the
following: is analytic or not at the origin
of the plane? (where the complex generalization of is assumed). This
question is also relevant in statistical mechanics because
, where is the ground state entropy of the
-state Potts antiferromagnet on the lattice graph , and the
analyticity of at is necessary for the large- series
expansions of . Although is analytic at for many
, there are some for which it is not; for these, has no
large- series expansion. It is important to understand the reason for this
nonanalyticity. Here we give a general condition that determines whether or not
a particular is analytic at and explains the
nonanalyticity where it occurs. We also construct infinite families of graphs
with functions that are non-analytic at and investigate the
properties of these functions. Our results are consistent with the conjecture
that a sufficient condition for to be analytic at is
that is a regular lattice graph . (This is known not to be a
necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in
Phys. Rev.
A Swendsen-Wang update algorithm for the Symanzik improved sigma model
We study a generalization of Swendsen-Wang algorithm suited for Potts models
with next-next-neighborhood interactions. Using the embedding technique
proposed by Wolff we test it on the Symanzik improved bidimensional non-linear
model. For some long range observables we find a little slowing down
exponent () that we interpret as an effect of the partial
frustration of the induced spin model.Comment: Self extracting archive fil
Combination of improved multibondic method and the Wang-Landau method
We propose a method for Monte Carlo simulation of statistical physical models
with discretized energy. The method is based on several ideas including the
cluster algorithm, the multicanonical Monte Carlo method and its acceleration
proposed recently by Wang and Landau. As in the multibondic ensemble method
proposed by Janke and Kappler, the present algorithm performs a random walk in
the space of the bond population to yield the state density as a function of
the bond number. A test on the Ising model shows that the number of Monte Carlo
sweeps required of the present method for obtaining the density of state with a
given accuracy is proportional to the system size, whereas it is proportional
to the system size squared for other conventional methods. In addition, the new
method shows a better performance than the original Wang-Landau method in
measurement of physical quantities.Comment: 12 pages, 3 figure
Crossover from Isotropic to Directed Percolation
Directed percolation is one of the generic universality classes for dynamic
processes. We study the crossover from isotropic to directed percolation by
representing the combined problem as a random cluster model, with a parameter
controlling the spontaneous birth of new forest fires. We obtain the exact
crossover exponent at using Coulomb gas methods in 2D.
Isotropic percolation is stable, as is confirmed by numerical finite-size
scaling results. For , the stability seems to change. An intuitive
argument, however, suggests that directed percolation at is unstable and
that the scaling properties of forest fires at intermediate values of are
in the same universality class as isotropic percolation, not only in 2D, but in
all dimensions.Comment: 4 pages, REVTeX, 4 epsf-emedded postscript figure
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