53 research outputs found
Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
Recently, we argued [Chin. Phys. Lett. , 080502 (2022)] that the Ising
model simultaneously exhibits two upper critical dimensions in
the Fortuin-Kasteleyn (FK) random-cluster representation. In this paper, we
perform a systematic study of the FK Ising model on hypercubic lattices with
spatial dimensions from 5 to 7, and on the complete graph. We provide a
detailed data analysis of the critical behaviors of a variety of quantities at
and near the critical points. Our results clearly show that many quantities
exhibit distinct critical phenomena for and , and thus
strongly support the argument that is also an upper critical dimension.
Moreover, for each studied dimension, we observe the existence of two
configuration sectors, two lengthscales, as well as two scaling windows, and
thus, two sets of critical exponents are needed to describe these behaviors.
Our finding enriches the understanding of the critical phenomena in the Ising
model.Comment: 17 pages, 17 figure
Geometric explanation of anomalous finite-size scaling in high dimensions
We give an intuitive geometric explanation for the apparent breakdown of
standard finite-size scaling in systems with periodic boundaries above the
upper critical dimension. The Ising model and self-avoiding walk are simulated
on five-dimensional hypercubic lattices with free and periodic boundary
conditions, by using geometric representations and recently introduced
Markov-chain Monte Carlo algorithms. We show that previously observed anomalous
behaviour for correlation functions, measured on the standard Euclidean scale,
can be removed by defining correlation functions on a scale which correctly
accounts for windings.Comment: 5 pages, 4 figure
Geometric properties of the complete-graph Ising model in the loop representation
The exact solution of the Ising model on the complete graph (CG) provides an
important, though mean-field, insight for the theory of continuous phase
transitions. Besides the original spin, the Ising model can be formulated in
the Fortuin-Kasteleyn random-cluster and the loop representation, in which many
geometric quantities have no correspondence in the spin representations. Using
a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop
representation, and, based on theoretical and numerical analyses, obtain a
number of exact results including volume fractal dimensions and scaling forms.
Moreover, by combining with the Loop-Cluster algorithm, we demonstrate how the
loop representation can provide an intuitive understanding to the recently
observed rich geometric phenomena in the random-cluster representation,
including the emergence of two configuration sectors, two length scales and two
scaling windows.Comment: 10 pages, 10 figure
Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension
Percolation has two mean-field theories, the Gaussian fixed point (GFP) and
the Landau mean-field theory or the complete graph (CG) asymptotics. By
large-scale Monte Carlo simulations, we systematically study the interplay of
the GFP and CG effects to the finite-size scaling of percolation above the
upper critical dimension with periodic, free, and cylindrical
boundary conditions. Our results suggest that, with periodic boundaries, the
\emph{unwrapped} correlation length scales as at the critical point,
diverging faster than above . As a consequence, the scaling behaviours
of macroscopic quantities with respect to the linear system size follow the
CG asymptotics. The distance-dependent properties, such as the short-distance
behaviour of the two-point correlation function and the Fourier transformed
quantities with non-zero modes, are still controlled by the GFP. With free
boundaries, since the correlation length is cutoff by , the finite-size
scaling at the critical point is controlled by the GFP. However, some
quantities are observed to exhibit the CG aysmptotics at the low-temperature
pseudo-critical point, such as the sizes of the two largest clusters. With
cylindrical boundaries, due to the interplay of the GFP and CG effects, the
correlation length along the axial direction of the cylinder scales as within the critical window of size ,
distinct from both periodic and free boundaries. A field-theoretical
calculation for deriving the scaling of is also presented. Moreover,
the one-point surface correlation function along the axial direction of the
cylinder is observed to scale as for short distance but then
enter a plateau of order before it decays significantly fast.Comment: 12 pages, 11 figure
High-precision Monte Carlo study of directed percolation in (d+1) dimensions
We present a Monte Carlo study of the bond and site directed (oriented)
percolation models in dimensions on simple-cubic and
body-centered-cubic lattices, with . A dimensionless ratio is
defined, and an analysis of its finite-size scaling produces improved estimates
of percolation thresholds. We also report improved estimates for the standard
critical exponents. In addition, we study the probability distributions of the
number of wet sites and radius of gyration, for .Comment: 11 pages, 21 figure
Shortest-Path Fractal Dimension for Percolation in Two and Three Dimensions
We carry out a high-precision Monte Carlo study of the shortest-path fractal
dimension \dm for percolation in two and three dimensions, using the
Leath-Alexandrowicz method which grows a cluster from an active seed site. A
variety of quantities are sampled as a function of the chemical distance,
including the number of activated sites, a measure of the radius, and the
survival probability. By finite-size scaling, we determine \dm = 1.130 77(2)
and in two and three dimensions, respectively. The result in 2D
rules out the recently conjectured value \dm=217/192 [Phys. Rev. E 81,
020102(R) (2010)].Comment: 5 pages, 4 figure
Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
Besides its original spin representation, the Ising model is known to have
the Fortuin-Kasteleyn (FK) bond and loop representations, of which the former
was recently shown to exhibit two upper critical dimensions .
Using a lifted worm algorithm, we determine the critical coupling as for , which significantly improves over the previous
results, and then study critical geometric properties of the loop-Ising
clusters on tori for spatial dimensions to 7. We show that, as the spin
representation, the loop Ising model has only one upper critical dimension at
. However, sophisticated finite-size scaling (FSS) behaviors, like two
length scales, two configuration sectors and two scaling windows, still exist
as the interplay effect of the Gaussian fixed point and complete-graph
asymptotics. Moreover, using the Loop-Cluster algorithm, we provide an
intuitive understanding of the emergence of the percolation-like upper critical
dimension in the FK-Ising model. As a consequence, a unified physical
picture is established for the FSS behaviors in all the three representations
of the Ising model above .Comment: 11 pages, 12 figure
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