114 research outputs found
Critical manifold of the kagome-lattice Potts model
Any two-dimensional infinite regular lattice G can be produced by tiling the
plane with a finite subgraph B of G; we call B a basis of G. We introduce a
two-parameter graph polynomial P_B(q,v) that depends on B and its embedding in
G. The algebraic curve P_B(q,v) = 0 is shown to provide an approximation to the
critical manifold of the q-state Potts model, with coupling v = exp(K)-1,
defined on G. This curve predicts the phase diagram both in the ferromagnetic
(v>0) and antiferromagnetic (v<0) regions. For larger bases B the
approximations become increasingly accurate, and we conjecture that P_B(q,v) =
0 provides the exact critical manifold in the limit of infinite B. Furthermore,
for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises
for any choice of B: the zero set of the recurrent factor then provides the
exact critical manifold. In this sense, the computation of P_B(q,v) can be used
to detect exact solvability of the Potts model on G.
We illustrate the method for the square lattice, where the Potts model has
been exactly solved, and the kagome lattice, where it has not. For the square
lattice we correctly reproduce the known phase diagram, including the
antiferromagnetic transition and the singularities in the Berker-Kadanoff
phase. For the kagome lattice, taking the smallest basis with six edges we
recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases
provide successive improvements on this formula, giving a natural extension of
Wu's approach. The polynomial predictions are in excellent agreement with
numerical computations. For v>0 the accuracy of the predicted critical coupling
v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to
10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).Comment: 31 pages, 12 figure
Exact bond percolation thresholds in two dimensions
Recent work in percolation has led to exact solutions for the site and bond
critical thresholds of many new lattices. Here we show how these results can be
extended to other classes of graphs, significantly increasing the number and
variety of solved problems. Any graph that can be decomposed into a certain
arrangement of triangles, which we call self-dual, gives a class of lattices
whose percolation thresholds can be found exactly by a recently introduced
triangle-triangle transformation. We use this method to generalize Wierman's
solution of the bow-tie lattice to yield several new solutions. We also give
another example of a self-dual arrangement of triangles that leads to a further
class of solvable problems. There are certainly many more such classes.Comment: Accepted for publication in J. Phys
Polynomial sequences for bond percolation critical thresholds
In this paper, I compute the inhomogeneous (multi-probability) bond critical
surfaces for the (4,6,12) and (3^4,6) lattices using the linearity
approximation described in (Scullard and Ziff, J. Stat. Mech. P03021),
implemented as a branching process of lattices. I find the estimates for the
bond percolation thresholds, p_c(4,6,12)=0.69377849... and
p_c(3^4,6)=0.43437077..., compared with Parviainen's numerical results of p_c
\approx 0.69373383 and p_c \approx 0.43430621 . These deviations are of the
order 10^{-5}, as is standard for this method, although they are outside
Parviainen's typical standard error of 10^{-7}. Deriving thresholds in this way
for a given lattice leads to a polynomial with integer coefficients, the root
in [0,1] of which gives the estimate for the bond threshold. I show how the
method can be refined, leading to a sequence of higher order polynomials making
predictions that likely converge to the exact answer. Finally, I discuss how
this fact hints that for certain graphs, such as the kagome lattice, the exact
bond threshold may not be the root of any polynomial with integer coefficients.Comment: submitted to Journal of Statistical Mechanic
Predictions of bond percolation thresholds for the kagom\'e and Archimedean lattices
Here we show how the recent exact determination of the bond percolation
threshold for the martini lattice can be used to provide approximations to the
unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one
of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2)
bond problems, and the other gives estimates of for the homogeneous
kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively
agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure
Critical surfaces for general inhomogeneous bond percolation problems
We present a method of general applicability for finding exact or accurate
approximations to bond percolation thresholds for a wide class of lattices. To
every lattice we sytematically associate a polynomial, the root of which in
is the conjectured critical point. The method makes the correct
prediction for every exactly solved problem, and comparison with numerical
results shows that it is very close, but not exact, for many others. We focus
primarily on the Archimedean lattices, in which all vertices are equivalent,
but this restriction is not crucial. Some results we find are kagome:
, , ,
, , :
. The results are generally within of numerical
estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in
the formulas (if they are not exact) are less than .Comment: Submitted to J. Stat. Mec
Exact Site Percolation Thresholds Using the Site-to-Bond and Star-Triangle Transformations
I construct a two-dimensional lattice on which the inhomogeneous site
percolation threshold is exactly calculable and use this result to find two
more lattices on which the site thresholds can be determined. The primary
lattice studied here, the ``martini lattice'', is a hexagonal lattice with
every second site transformed into a triangle. The site threshold of this
lattice is found to be , while the others have and
. This last solution suggests a possible approach to establishing
the bound for the hexagonal site threshold, . To derive these
results, I solve a correlated bond problem on the hexagonal lattice by use of
the star-triangle transformation and then, by a particular choice of
correlations, solve the site problem on the martini lattice.Comment: 12 pages, 10 figures. Submitted to Physical Review
The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
We give a conditional derivation of the inhomogeneous critical percolation manifold of the bow-tie lattice with five different probabilities, a problem that does not appear at first to fall into any known solvable class. Although our argument is mathematically rigorous only on a region of the manifold, we conjecture that the formula is correct over its entire domain, and we provide a non-rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface. We discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow-tie lattices, and that, if it could be put on a firm foundation, the negative probability portion of our method would extend this class to many further systems, including F Y Wu’s checkerboard formula for the square lattice. We conclude by showing that this latter problem can in fact be proved using a recent result of Grimmett and Manolescu for isoradial graphs, lending strong evidence in favor of our other conjectured results. This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/98528/1/1751-8121_45_49_494005.pd
Testing refinements by refining tests
One of the potential benefits of formal methods is that they offer the possibility of reducing the costs of testing. A specification acts as both the benchmark against which any implementation is tested, and also as the means by which tests are generated. There has therefore been interest in developing test generation techniques from formal specifications, and a number of different methods have been derived for state based languages such as Z, B and VDM. However, in addition to deriving tests from a formal specification, we might wish to refine the specification further before its implementation. The purpose of this paper is to explore the relationship between testing and refinement. As our model for test generation we use a DNF partition analysis for operations written in Z, which produces a number of disjoint test cases for each operation. In this paper we discuss how the partition analysis of an operation alters upon refinement, and we develop techniques that allow us to refine abstract tests in order to generate test cases for a refinement. To do so we use (and extend existing) methods for calculating the weakest data refinement of a specification
Classical phase transitions in a one-dimensional short-range spin model
Ising's solution of a classical spin model famously demonstrated the absence
of a positive-temperature phase transition in one-dimensional equilibrium
systems with short-range interactions. No-go arguments established that the
energy cost to insert domain walls in such systems is outweighed by entropy
excess so that symmetry cannot be spontaneously broken. An archetypal way
around the no-go theorems is to augment interaction energy by increasing the
range of interaction. Here we introduce new ways around the no-go theorems by
investigating entropy depletion instead. We implement this for the Potts model
with invisible states.Because spins in such a state do not interact with their
surroundings, they contribute to the entropy but not the interaction energy of
the system. Reducing the number of invisible states to a negative value
decreases the entropy by an amount sufficient to induce a positive-temperature
classical phase transition. This approach is complementary to the long-range
interaction mechanism. Alternatively, subjecting positive numbers of invisible
states to imaginary or complex fields can trigger such a phase transition. We
also discuss potential physical realisability of such systems.Comment: 29 pages, 11 figure
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