306 research outputs found
No directed fractal percolation in zero area
We show that fractal (or "Mandelbrot") percolation in two dimensions produces
a set containing no directed paths, when the set produced has zero area. This
improves a similar result by the first author in the case of constant retention
probabilities to the case of retention probabilities approaching 1
Vertex Models and Random Labyrinths: Phase Diagrams for Ice-type Vertex Models
We propose a simple geometric recipe for constructing phase diagrams for a
general class of vertex models obeying the ice rule. The disordered phase maps
onto the intersecting loop model which is interesting in its own right and is
related to several other statistical mechanical models. This mapping is also
useful in understanding some ordered phases of these vertex models as they
correspond to the polymer loop models with cross-links in their vulcanised
phase.Comment: 8 pages, 6 figure
Cluster Monte Carlo study of multi-component fluids of the Stillinger-Helfand and Widom-Rowlinson type
Phase transitions of fluid mixtures of the type introduced by Stillinger and
Helfand are studied using a continuum version of the invaded cluster algorithm.
Particles of the same species do not interact, but particles of different types
interact with each other via a repulsive potential. Examples of interactions
include the Gaussian molecule potential and a repulsive step potential.
Accurate values of the critical density, fugacity and magnetic exponent are
found in two and three dimensions for the two-species model. The effect of
varying the number of species and of introducing quenched impurities is also
investigated. In all the cases studied, mixtures of -species are found to
have properties similar to -state Potts models.Comment: 25 pages, 5 figure
Monte Carlo study of the Widom-Rowlinson fluid using cluster methods
The Widom-Rowlinson model of a fluid mixture is studied using a new cluster
algorithm that is a generalization of the invaded cluster algorithm previously
applied to Potts models. Our estimate of the critical exponents for the
two-component fluid are consistent with the Ising universality class in two and
three dimensions. We also present results for the three-component fluid.Comment: 13 pages RevTex and 2 Postscript figure
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
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
Avoided Critical Behavior in O(n) Systems
Long-range frustrating interactions, even if their strength is infinitesimal,
can give rise to a dramatic proliferations of ground or near-ground states. As
a consequence, the ordering temperature can exhibit a discontinuous drop as a
function of the frustration. A simple model of the doped Mott insulator, where
the short-range tendency of the holes to phase separate competes with
long-range Coulomb effects, exhibits this "avoided critical" behavior. This
model may serve as a paradigm for many other systems.Comment: 4 pages, 2 figure
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
Universal Spectral Correlations at the Mobility Edge
We demonstrate the level statistics in the vicinity of the Anderson
transition in dimensions to be universal and drastically different from
both Wigner-Dyson in the metallic regime and Poisson in the insulator regime.
The variance of the number of levels in a given energy interval with
is proved to behave as where
and is the correlation length exponent. The
inequality , shown to be required by an exact sum rule, results from
nontrivial cancellations (due to the causality and scaling requirements) in
calculating the two-level correlation function.Comment: REVTeX, 12pages, +1 postscript figure (included
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
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