412 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
Ensemble dependence in the Random transverse-field Ising chain
In a disordered system one can either consider a microcanonical ensemble,
where there is a precise constraint on the random variables, or a canonical
ensemble where the variables are chosen according to a distribution without
constraints. We address the question as to whether critical exponents in these
two cases can differ through a detailed study of the random transverse-field
Ising chain. We find that the exponents are the same in both ensembles, though
some critical amplitudes vanish in the microcanonical ensemble for correlations
which span the whole system and are particularly sensitive to the constraint.
This can \textit{appear} as a different exponent. We expect that this apparent
dependence of exponents on ensemble is related to the integrability of the
model, and would not occur in non-integrable models.Comment: 8 pages, 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
Disorder Averaging and Finite Size Scaling
We propose a new picture of the renormalization group (RG) approach in the
presence of disorder, which considers the RG trajectories of each random sample
(realization) separately instead of the usual renormalization of the averaged
free energy. The main consequence of the theory is that the average over
randomness has to be taken after finding the critical point of each
realization. To demonstrate these concepts, we study the finite-size scaling
properties of the two-dimensional random-bond Ising model. We find that most of
the previously observed finite-size corrections are due to the sample-to-sample
fluctuation of the critical temperature and scaling is more adequate in terms
of the new scaling variables.Comment: 4 pages, 6 figures include
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
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
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
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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