289 research outputs found
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
New Algorithm of the Finite Lattice Method for the High-temperature Expansion of the Ising Model in Three Dimensions
We propose a new algorithm of the finite lattice method to generate the
high-temperature series for the Ising model in three dimensions. It enables us
to extend the series for the free energy of the simple cubic lattice from the
previous series of 26th order to 46th order in the inverse temperature. The
obtained series give the estimate of the critical exponent for the specific
heat in high precision.Comment: 4 pages, 4 figures, submitted to Phys. Rev. Letter
Large- expansion of the specific heat for the two-dimensional -state Potts model
We have calculated the large- expansion for the specific heat at the phase
transition point in the two-dimensional -state Potts model to the 23rd order
in using the finite lattice method. The obtained series allows us
to give highly convergent estimates of the specific heat for on the first
order transition point. The result confirm us the correctness of the conjecture
by Bhattacharya et al. on the asymptotic behavior of the specific heat for .Comment: 7 pages, LaTeX, 2 postscript 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
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
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
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
Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails
We give exact relations for a number of amplitude combinations that occur in
the study of self-avoiding walks, polygons and lattice trails. In particular,
we elucidate the lattice-dependent factors which occur in those combinations
which are otherwise universal, show how these are modified for oriented
lattices, and give new results for amplitude ratios involving even moments of
the area of polygons. We also survey numerical results for a wide range of
amplitudes on a number of oriented and regular lattices, and provide some new
ones.Comment: 20 pages, NI 92016, OUTP 92-54S, UCSBTH-92-5
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