1,075 research outputs found
A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice
We present a new and more efficient implementation of transfer-matrix methods
for exact enumerations of lattice objects. The new method is illustrated by an
application to the enumeration of self-avoiding polygons on the square lattice.
A detailed comparison with the previous best algorithm shows significant
improvement in the running time of the algorithm. The new algorithm is used to
extend the enumeration of polygons to length 130 from the previous record of
110.Comment: 17 pages, 8 figures, IoP style file
Series expansions of the percolation probability on the directed triangular lattice
We have derived long series expansions of the percolation probability for
site, bond and site-bond percolation on the directed triangular lattice. For
the bond problem we have extended the series from order 12 to 51 and for the
site problem from order 12 to 35. For the site-bond problem, which has not been
studied before, we have derived the series to order 32. Our estimates of the
critical exponent are in full agreement with results for similar
problems on the square lattice, confirming expectations of universality. For
the critical probability and exponent we find in the site case: and ; in the bond case:
and ; and in the site-bond
case: and . In
addition we have obtained accurate estimates for the critical amplitudes. In
all cases we find that the leading correction to scaling term is analytic,
i.e., the confluent exponent .Comment: 26 pages, LaTeX. To appear in J. Phys.
Directed percolation near a wall
Series expansion methods are used to study directed bond percolation clusters
on the square lattice whose lateral growth is restricted by a wall parallel to
the growth direction. The percolation threshold is found to be the same
as that for the bulk. However the values of the critical exponents for the
percolation probability and mean cluster size are quite different from those
for the bulk and are estimated by and respectively. On the other hand the exponent
characterising the scale of the cluster size
distribution is found to be unchanged by the presence of the wall.
The parallel connectedness length, which is the scale for the cluster length
distribution, has an exponent which we estimate to be and is also unchanged. The exponent of the mean
cluster length is related to and by the scaling
relation and using the above estimates
yields to within the accuracy of our results. We conjecture that
this value of is exact and further support for the conjecture is
provided by the direct series expansion estimate .Comment: 12pages LaTeX, ioplppt.sty, to appear in J. Phys.
Complex-Temperature Singularities in the Ising Model. III. Honeycomb Lattice
We study complex-temperature properties of the uniform and staggered
susceptibilities and of the Ising model on the honeycomb
lattice. From an analysis of low-temperature series expansions, we find
evidence that and both have divergent singularities at the
point (where ), with exponents
. The critical amplitudes at this
singularity are calculated. Using exact results, we extract the behaviour of
the magnetisation and specific heat at complex-temperature
singularities. We find that, in addition to its zero at the physical critical
point, diverges at with exponent , vanishes
continuously at with exponent , and vanishes
discontinuously elsewhere along the boundary of the complex-temperature
ferromagnetic phase. diverges at with exponent
and at (where ) with exponent , and
diverges logarithmically at . We find that the exponent relation
is violated at ; the right-hand side is 4
rather than 2. The connections of these results with complex-temperature
properties of the Ising model on the triangular lattice are discussed.Comment: 22 pages, latex, figures appended after the end of the text as a
compressed, uuencoded postscript fil
Partially directed paths in a wedge
The enumeration of lattice paths in wedges poses unique mathematical
challenges. These models are not translationally invariant, and the absence of
this symmetry complicates both the derivation of a functional recurrence for
the generating function, and solving for it. In this paper we consider a model
of partially directed walks from the origin in the square lattice confined to
both a symmetric wedge defined by , and an asymmetric wedge defined
by the lines and Y=0, where is an integer. We prove that the
growth constant for all these models is equal to , independent of
the angle of the wedge. We derive functional recursions for both models, and
obtain explicit expressions for the generating functions when . From these
we find asymptotic formulas for the number of partially directed paths of
length in a wedge when .
The functional recurrences are solved by a variation of the kernel method,
which we call the ``iterated kernel method''. This method appears to be similar
to the obstinate kernel method used by Bousquet-Melou. This method requires us
to consider iterated compositions of the roots of the kernel. These
compositions turn out to be surprisingly tractable, and we are able to find
simple explicit expressions for them. However, in spite of this, the generating
functions turn out to be similar in form to Jacobi -functions, and have
natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT
Force induced triple point for interacting polymers
We show the existence of a force induced triple point in an interacting
polymer problem that allows two zero-force thermal phase transitions. The phase
diagrams for three different models of mutually attracting but self avoiding
polymers are presented. One of these models has an intermediate phase and it
shows a triple point but not the others. A general phase diagram with
multicritical points in an extended parameter space is also discussed.Comment: 4 pages, 8 figures, revtex
Unexpected Spin-Off from Quantum Gravity
We propose a novel way of investigating the universal properties of spin
systems by coupling them to an ensemble of causal dynamically triangulated
lattices, instead of studying them on a fixed regular or random lattice.
Somewhat surprisingly, graph-counting methods to extract high- or
low-temperature series expansions can be adapted to this case. For the
two-dimensional Ising model, we present evidence that this ameliorates the
singularity structure of thermodynamic functions in the complex plane, and
improves the convergence of the power series.Comment: 10 pages, 4 figures; final, slightly amended version, to appear in
Physica
Self-avoiding walks crossing a square
We study a restricted class of self-avoiding walks (SAW) which start at the
origin (0, 0), end at , and are entirely contained in the square on the square lattice . The number of distinct
walks is known to grow as . We estimate as well as obtaining strict upper and lower bounds,
We give exact results for the number of SAW of
length for and asymptotic results for .
We also consider the model in which a weight or {\em fugacity} is
associated with each step of the walk. This gives rise to a canonical model of
a phase transition. For the average length of a SAW grows as ,
while for it grows as
. Here is the growth constant of unconstrained SAW in . For we provide numerical evidence, but no proof, that the
average walk length grows as .
We also consider Hamiltonian walks under the same restriction. They are known
to grow as on the same lattice. We give
precise estimates for as well as upper and lower bounds, and prove that
Comment: 27 pages, 9 figures. Paper updated and reorganised following
refereein
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
Effects of Eye-phase in DNA unzipping
The onset of an "eye-phase" and its role during the DNA unzipping is studied
when a force is applied to the interior of the chain. The directionality of the
hydrogen bond introduced here shows oscillations in force-extension curve
similar to a "saw-tooth" kind of oscillations seen in the protein unfolding
experiments. The effects of intermediates (hairpins) and stacking energies on
the melting profile have also been discussed.Comment: RevTeX v4, 9 pages with 7 eps figure
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