1,602 research outputs found

### 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 $Y = \pm pX$, and an asymmetric wedge defined
by the lines $Y= pX$ and Y=0, where $p > 0$ is an integer. We prove that the
growth constant for all these models is equal to $1+\sqrt{2}$, independent of
the angle of the wedge. We derive functional recursions for both models, and
obtain explicit expressions for the generating functions when $p=1$. From these
we find asymptotic formulas for the number of partially directed paths of
length $n$ in a wedge when $p=1$.
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 $\theta$-functions, and have
natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT

### Series studies of the Potts model. I: The simple cubic Ising model

The finite lattice method of series expansion is generalised to the $q$-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 ($q=2$) case we have extended low-temperature series for the
partition functions, magnetisation and zero-field susceptibility to $u^{26}$
from $u^{20}$. The high-temperature series for the zero-field partition
function is extended from $v^{18}$ to $v^{22}$. 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

### Complex-Temperature Singularities in the $d=2$ Ising Model. III. Honeycomb Lattice

We study complex-temperature properties of the uniform and staggered
susceptibilities $\chi$ and $\chi^{(a)}$ of the Ising model on the honeycomb
lattice. From an analysis of low-temperature series expansions, we find
evidence that $\chi$ and $\chi^{(a)}$ both have divergent singularities at the
point $z=-1 \equiv z_{\ell}$ (where $z=e^{-2K}$), with exponents
$\gamma_{\ell}'= \gamma_{\ell,a}'=5/2$. The critical amplitudes at this
singularity are calculated. Using exact results, we extract the behaviour of
the magnetisation $M$ and specific heat $C$ at complex-temperature
singularities. We find that, in addition to its zero at the physical critical
point, $M$ diverges at $z=-1$ with exponent $\beta_{\ell}=-1/4$, vanishes
continuously at $z=\pm i$ with exponent $\beta_s=3/8$, and vanishes
discontinuously elsewhere along the boundary of the complex-temperature
ferromagnetic phase. $C$ diverges at $z=-1$ with exponent $\alpha_{\ell}'=2$
and at $v=\pm i/\sqrt{3}$ (where $v = \tanh K$) with exponent $\alpha_e=1$, and
diverges logarithmically at $z=\pm i$. We find that the exponent relation
$\alpha'+2\beta+\gamma'=2$ is violated at $z=-1$; 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

### 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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