280 research outputs found
Series studies of the Potts model. I: The simple cubic Ising model
The finite lattice method of series expansion is generalised to the -state
Potts model on the simple cubic lattice.
It is found that the computational effort grows exponentially with the square
of the number of series terms obtained, unlike two-dimensional lattices where
the computational requirements grow exponentially with the number of terms. For
the Ising () case we have extended low-temperature series for the
partition functions, magnetisation and zero-field susceptibility to
from . The high-temperature series for the zero-field partition
function is extended from to . Subsequent analysis gives
critical exponents in agreement with those from field theory.Comment: submitted to J. Phys. A: Math. Gen. Uses preprint.sty: included. 24
page
Osculating and neighbour-avoiding polygons on the square lattice
We study two simple modifications of self-avoiding polygons. Osculating
polygons are a super-set in which we allow the perimeter of the polygon to
touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest
neighbour vertices provided these are joined by the associated edge and thus
form a sub-set of self-avoiding polygons. We use the finite lattice method to
count the number of osculating polygons and neighbour-avoiding polygons on the
square lattice. We also calculate their radius of gyration and the first
area-weighted moment. Analysis of the series confirms exact predictions for the
critical exponents and the universality of various amplitude combinations. For
both cases we have found exact solutions for the number of convex and
almost-convex polygons.Comment: 14 pages, 5 figure
Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
We present an efficient algorithm for computing the partition function of the
q-colouring problem (chromatic polynomial) on regular two-dimensional lattice
strips. Our construction involves writing the transfer matrix as a product of
sparse matrices, each of dimension ~ 3^m, where m is the number of lattice
spacings across the strip. As a specific application, we obtain the large-q
series of the bulk, surface and corner free energies of the chromatic
polynomial. This extends the existing series for the square lattice by 32
terms, to order q^{-79}. On the triangular lattice, we verify Baxter's
analytical expression for the bulk free energy (to order q^{-40}), and we are
able to conjecture exact product formulae for the surface and corner free
energies.Comment: 17 pages. Version 2: added 4 further term to the serie
Size and area of square lattice polygons
We use the finite lattice method to calculate the radius of gyration, the
first and second area-weighted moments of self-avoiding polygons on the square
lattice. The series have been calculated for polygons up to perimeter 82.
Analysis of the series yields high accuracy estimates confirming theoretical
predictions for the value of the size exponent, , and certain
universal amplitude combinations. Furthermore, a detailed analysis of the
asymptotic form of the series coefficients provide the firmest evidence to date
for the existence of a correction-to-scaling exponent, .Comment: 12 pages 3 figure
A parallel algorithm for the enumeration of benzenoid hydrocarbons
We present an improved parallel algorithm for the enumeration of fixed
benzenoids B_h containing h hexagonal cells. We can thus extend the enumeration
of B_h from the previous best h=35 up to h=50. Analysis of the associated
generating function confirms to a very high degree of certainty that and we estimate that the growth constant and the amplitude .Comment: 14 pages, 6 figure
Punctured polygons and polyominoes on the square lattice
We use the finite lattice method to count the number of punctured staircase
and self-avoiding polygons with up to three holes on the square lattice. New or
radically extended series have been derived for both the perimeter and area
generating functions. We show that the critical point is unchanged by a finite
number of punctures, and that the critical exponent increases by a fixed amount
for each puncture. The increase is 1.5 per puncture when enumerating by
perimeter and 1.0 when enumerating by area. A refined estimate of the
connective constant for polygons by area is given. A similar set of results is
obtained for finitely punctured polyominoes. The exponent increase is proved to
be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure
Series studies of the Potts model. II: Bulk series for the square lattice
The finite lattice method of series expansion has been used to extend
low-temperature series for the partition function, order parameter and
susceptibility of the -state Potts model to order (i.e. ),
, , , , , , and
for , 3, 4, \dots 9 and 10 respectively. These series are used
to test techniques designed to distinguish first-order transitions from
continuous transitions. New numerical values are also obtained for the
-state Potts model with .Comment: 32 pages, incl. 3 figures, incl. 3 figure
Scaling prediction for self-avoiding polygons revisited
We analyse new exact enumeration data for self-avoiding polygons, counted by
perimeter and area on the square, triangular and hexagonal lattices. In
extending earlier analyses, we focus on the perimeter moments in the vicinity
of the bicritical point. We also consider the shape of the critical curve near
the bicritical point, which describes the crossover to the branched polymer
phase. Our recently conjectured expression for the scaling function of rooted
self-avoiding polygons is further supported. For (unrooted) self-avoiding
polygons, the analysis reveals the presence of an additional additive term with
a new universal amplitude. We conjecture the exact value of this amplitude.Comment: 17 pages, 3 figure
Test of Guttmann and Enting's conjecture in the eight-vertex model
We investigate the analyticity property of the partially resummed series
expansion(PRSE) of the partition function for the eight-vertex model.
Developing a graphical technique, we have obtained a first few terms of the
PRSE and found that these terms have a pole only at one point in the complex
plane of the coupling constant. This result supports the conjecture proposed by
Guttmann and Enting concerning the ``solvability'' in statistical mechanical
lattice models.Comment: 15 pages, 3 figures, RevTe
Model for Anisotropic Directed Percolation
We propose a simulation model to study the properties of directed percolation
in two-dimensional (2D) anisotropic random media. The degree of anisotropy in
the model is given by the ratio between the axes of a semi-ellipse
enclosing the bonds that promote percolation in one direction. At percolation,
this simple model shows that the average number of bonds per site in 2D is an
invariant equal to 2.8 independently of . This result suggests that
Sinai's theorem proposed originally for isotropic percolation is also valid for
anisotropic directed percolation problems. The new invariant also yields a
constant fractal dimension for all , which is the same
value found in isotropic directed percolation (i.e., ).Comment: RevTeX, 9 pages, 3 figures. To appear in Phys.Rev.
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