138 research outputs found
An Importance Sampling Algorithm for the Ising Model with Strong Couplings
We consider the problem of estimating the partition function of the
ferromagnetic Ising model in a consistent external magnetic field. The
estimation is done via importance sampling in the dual of the Forney factor
graph representing the model. Emphasis is on models at low temperature
(corresponding to models with strong couplings) and on models with a mixture of
strong and weak coupling parameters.Comment: Proc. 2016 Int. Zurich Seminar on Communications (IZS), Zurich,
Switzerland, March 2-4, 2016, pp. 180-18
Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
We present exact calculations of the partition function of the -state
Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the
honeycomb (brick) lattice of width and arbitrarily great length. In the
infinite-length limit the thermodynamic properties are discussed. The
continuous locus of singularities of the free energy is determined in the
plane for fixed temperature and in the complex temperature plane for fixed
values. We also give exact calculations of the zero-temperature partition
function (chromatic polynomial) and , the exponent of the ground-state
entropy, for the Potts antiferromagnet for honeycomb strips of type (iv)
, cyclic, (v) , M\"obius, (vi) , cylindrical, and (vii)
, open. In the infinite-length limit we calculate and determine
the continuous locus of points where it is nonanalytic. We show that our exact
calculation of the entropy for the strip with cylindrical boundary
conditions provides an extremely accurate approximation, to a few parts in
for moderate values, to the entropy for the full 2D honeycomb
lattice (where the latter is determined by Monte Carlo measurements since no
exact analytic form is known).Comment: 48 pages, latex, with encapsulated postscript figure
Dynamic Critical Behavior of a Swendsen-Wang-Type Algorithm for the Ashkin-Teller Model
We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for
the Ashkin--Teller model. We find that the Li--Sokal bound on the
autocorrelation time ()
holds along the self-dual curve of the symmetric Ashkin--Teller model, and is
almost but not quite sharp. The ratio appears
to tend to infinity either as a logarithm or as a small power (). In an appendix we discuss the problem of extracting estimates of
the exponential autocorrelation time.Comment: 59 pages including 3 figures, uuencoded g-compressed ps file.
Postscript size = 799740 byte
Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
We conjecture an exact form for an universal ratio of four-point cluster
connectivities in the critical two-dimensional -color Potts model. We also
provide analogous results for the limit that corresponds to
percolation where the observable has a logarithmic singularity. Our conjectures
are tested against Monte Carlo simulations showing excellent agreement for
.Comment: 29 pages, 9 Figures. Published version: improved discussion,
additional numerical tests and reference
A critical Ising model on the Labyrinth
A zero-field Ising model with ferromagnetic coupling constants on the
so-called Labyrinth tiling is investigated. Alternatively, this can be regarded
as an Ising model on a square lattice with a quasi-periodic distribution of up
to eight different coupling constants. The duality transformation on this
tiling is considered and the self-dual couplings are determined. Furthermore,
we analyze the subclass of exactly solvable models in detail parametrizing the
coupling constants in terms of four rapidity parameters. For those, the
self-dual couplings correspond to the critical points which, as expected,
belong to the Onsager universality class.Comment: 25 pages, 6 figure
Thin Fisher Zeroes
Biskup et al. [Phys. Rev. Lett. 84 (2000) 4794] have recently suggested that
the loci of partition function zeroes can profitably be regarded as phase
boundaries in the complex temperature or field planes.
We obtain the Fisher zeroes for Ising and Potts models on non-planar
(``thin'') regular random graphs using this approach, and note that the locus
of Fisher zeroes on a Bethe lattice is identical to the corresponding random
graph. Since the number of states appears as a parameter in the Potts solution
the limiting locus of chromatic zeroes is also accessible.Comment: 10 pages, 4 figure
Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models
We demonstrate that the invaded cluster algorithm, recently introduced by
Machta et al, is a fast and reliable tool for determining the critical
temperature and the magnetic critical exponent of periodic and aperiodic
ferromagnetic Ising models in two dimensions. The algorithm is shown to
reproduce the known values of the critical temperature on various periodic and
quasiperiodic graphs with an accuracy of more than three significant digits. On
two quasiperiodic graphs which were not investigated in this respect before,
the twelvefold symmetric square-triangle tiling and the tenfold symmetric
T\"ubingen triangle tiling, we determine the critical temperature. Furthermore,
a generalization of the algorithm to non-identical coupling strengths is
presented and applied to a class of Ising models on the Labyrinth tiling. For
generic cases in which the heuristic Harris-Luck criterion predicts deviations
from the Onsager universality class, we find a magnetic critical exponent
different from the Onsager value. But also notable exceptions to the criterion
are found which consist not only of the exactly solvable cases, in agreement
with a recent exact result, but also of the self-dual ones and maybe more.Comment: 15 pages, 5 figures; v2: Fig. 5b replaced, minor change
Susceptibility amplitude ratios in the two-dimensional Potts model and percolation
The high-temperature susceptibility of the -state Potts model behaves as
as , while for one may define
both longitudinal and transverse susceptibilities, with the same power law but
different amplitudes and . We extend a previous analytic
calculation of the universal ratio in two dimensions to the
low-temperature ratio , and test both predictions with Monte
Carlo simulations for and 4. The data for are inconclusive owing to
large corrections to scaling, while for they appear consistent with the
prediction for , but not with that for . A
simple extrapolation of our analytic results to indicates a similar
discrepancy with the corresponding measured quantities in percolation. We point
out that stronger assumptions were made in the derivation of the ratio
, and our work suggests that these may be unjustified.Comment: 17 pages, late
- …