1,602 research outputs found

    Partially directed paths in a wedge

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    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=±pXY = \pm pX, and an asymmetric wedge defined by the lines Y=pXY= pX and Y=0, where p>0p > 0 is an integer. We prove that the growth constant for all these models is equal to 1+21+\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=1p=1. From these we find asymptotic formulas for the number of partially directed paths of length nn in a wedge when p=1p=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 θ\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

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    The finite lattice method of series expansion is generalised to the qq-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=2q=2) case we have extended low-temperature series for the partition functions, magnetisation and zero-field susceptibility to u26u^{26} from u20u^{20}. The high-temperature series for the zero-field partition function is extended from v18v^{18} to v22v^{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=2d=2 Ising Model. III. Honeycomb Lattice

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    We study complex-temperature properties of the uniform and staggered susceptibilities χ\chi and χ(a)\chi^{(a)} of the Ising model on the honeycomb lattice. From an analysis of low-temperature series expansions, we find evidence that χ\chi and χ(a)\chi^{(a)} both have divergent singularities at the point z=1zz=-1 \equiv z_{\ell} (where z=e2Kz=e^{-2K}), with exponents γ=γ,a=5/2\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 MM and specific heat CC at complex-temperature singularities. We find that, in addition to its zero at the physical critical point, MM diverges at z=1z=-1 with exponent β=1/4\beta_{\ell}=-1/4, vanishes continuously at z=±iz=\pm i with exponent βs=3/8\beta_s=3/8, and vanishes discontinuously elsewhere along the boundary of the complex-temperature ferromagnetic phase. CC diverges at z=1z=-1 with exponent α=2\alpha_{\ell}'=2 and at v=±i/3v=\pm i/\sqrt{3} (where v=tanhKv = \tanh K) with exponent αe=1\alpha_e=1, and diverges logarithmically at z=±iz=\pm i. We find that the exponent relation α+2β+γ=2\alpha'+2\beta+\gamma'=2 is violated at z=1z=-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

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    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