297 research outputs found
Cardy's Formula for Certain Models of the Bond-Triangular Type
We introduce and study a family of 2D percolation systems which are based on
the bond percolation model of the triangular lattice. The system under study
has local correlations, however, bonds separated by a few lattice spacings act
independently of one another. By avoiding explicit use of microscopic paths, it
is first established that the model possesses the typical attributes which are
indicative of critical behavior in 2D percolation problems. Subsequently, the
so called Cardy-Carleson functions are demonstrated to satisfy, in the
continuum limit, Cardy's formula for crossing probabilities. This extends the
results of S. Smirnov to a non-trivial class of critical 2D percolation
systems.Comment: 49 pages, 7 figure
Dynamic and static properties of the invaded cluster algorithm
Simulations of the two-dimensional Ising and 3-state Potts models at their
critical points are performed using the invaded cluster (IC) algorithm. It is
argued that observables measured on a sub-lattice of size l should exhibit a
crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the
lattice size L, and a scaling form is proposed to describe the crossover
phenomenon. It is found that the energy autocorrelation time tau(l,L) for an
l*l sub-lattice attains a maximum in the crossover region, and a dynamic
exponent z for the IC algorithm is defined according to tau_max ~ L^z.
Simulation results for the 3-state model yield z=.346(.002) which is smaller
than values of the dynamic exponent found for the SW and Wolff algorithms and
also less than the Li-Sokal bound. The results are less conclusive for the
Ising model, but it appears that z<.21 and possibly that tau_max ~ log L so
that z=0 -- similar to previous results for the SW and Wolff algorithms.Comment: 21 pages with 12 figure
Rejoinder to the Response arXiv:0812.2330 to 'Comment on a recent conjectured solution of the three-dimensional Ising model'
We comment on Z. D. Zhang's Response [arXiv:0812.2330] to our recent Comment
[arXiv:0811.3876] addressing the conjectured solution of the three-dimensional
Ising model reported in [arXiv:0705.1045].Comment: 2 page
Invaded cluster simulations of the XY model in two and three dimensions
The invaded cluster algorithm is used to study the XY model in two and three
dimensions up to sizes 2000^2 and 120^3 respectively. A soft spin O(2) model,
in the same universality class as the 3D XY model, is also studied. The static
critical properties of the model and the dynamical properties of the algorithm
are reported. The results are K_c=0.45412(2) for the 3D XY model and
eta=0.037(2) for the 3D XY universality class. For the 2D XY model the results
are K_c=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show
any critical slowing for the magnetization or critical temperature estimator
for the 2D or 3D XY models.Comment: 30 pages, 11 figures, problem viewing figures corrected in v
Orbital order in classical models of transition-metal compounds
We study the classical 120-degree and related orbital models. These are the
classical limits of quantum models which describe the interactions among
orbitals of transition-metal compounds. We demonstrate that at low temperatures
these models exhibit a long-range order which arises via an "order by disorder"
mechanism. This strongly indicates that there is orbital ordering in the
quantum version of these models, notwithstanding recent rigorous results on the
absence of spin order in these systems.Comment: 7 pages, 1 eps fi
Surface tension in the dilute Ising model. The Wulff construction
We study the surface tension and the phenomenon of phase coexistence for the
Ising model on \mathbbm{Z}^d () with ferromagnetic but random
couplings. We prove the convergence in probability (with respect to random
couplings) of surface tension and analyze its large deviations : upper
deviations occur at volume order while lower deviations occur at surface order.
We study the asymptotics of surface tension at low temperatures and relate the
quenched value of surface tension to maximal flows (first passage
times if ). For a broad class of distributions of the couplings we show
that the inequality -- where is the surface
tension under the averaged Gibbs measure -- is strict at low temperatures. We
also describe the phenomenon of phase coexistence in the dilute Ising model and
discuss some of the consequences of the media randomness. All of our results
hold as well for the dilute Potts and random cluster models
Graphical representations and cluster algorithms for critical points with fields
A two-replica graphical representation and associated cluster algorithm is
described that is applicable to ferromagnetic Ising systems with arbitrary
fields. Critical points are associated with the percolation threshold of the
graphical representation. Results from numerical simulations of the Ising model
in a staggered field are presented. The dynamic exponent for the algorithm is
measured to be less than 0.5.Comment: Revtex, 12 pages with 2 figure
Random Cluster Models on the Triangular Lattice
We study percolation and the random cluster model on the triangular lattice
with 3-body interactions. Starting with percolation, we generalize the
star--triangle transformation: We introduce a new parameter (the 3-body term)
and identify configurations on the triangles solely by their connectivity. In
this new setup, necessary and sufficient conditions are found for positive
correlations and this is used to establish regions of percolation and
non-percolation. Next we apply this set of ideas to the random cluster
model: We derive duality relations for the suitable random cluster measures,
prove necessary and sufficient conditions for them to have positive
correlations, and finally prove some rigorous theorems concerning phase
transitions.Comment: 24 pages, 1 figur
The Computational Complexity of Generating Random Fractals
In this paper we examine a number of models that generate random fractals.
The models are studied using the tools of computational complexity theory from
the perspective of parallel computation. Diffusion limited aggregation and
several widely used algorithms for equilibrating the Ising model are shown to
be highly sequential; it is unlikely they can be simulated efficiently in
parallel. This is in contrast to Mandelbrot percolation that can be simulated
in constant parallel time. Our research helps shed light on the intrinsic
complexity of these models relative to each other and to different growth
processes that have been recently studied using complexity theory. In addition,
the results may serve as a guide to simulation physics.Comment: 28 pages, LATEX, 8 Postscript figures available from
[email protected]
Comments on Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation
We compare the correlation times of the Sweeny and Gliozzi dynamics for
two-dimensional Ising and three-state Potts models, and the three-dimensional
Ising model for the simulations in the percolation prepresentation. The results
are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found
that Sweeny and Gliozzi dynamics have essentially the same dynamical critical
behavior. Contrary to Gliozzi's claim (cond-mat/0201285), the Gliozzi dynamics
has critical slowing down comparable to that of other cluster methods. For the
two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to
logarithmic size dependences; for two-dimensional three-state Potts model,
their dynamical critical exponent z is 0.49(1); the three-dimensional Ising
model has z = 0.37(2).Comment: RevTeX, 4 pages, 5 figure
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