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

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 $p_c$ 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 $\beta_1 = 0.7338 \pm 0.0001$ and $\gamma_1 = 1.8207 \pm 0.0004$ respectively. On the other hand the exponent $\Delta_1=\beta_1 +\gamma_1$ 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 $\nu_{1\parallel} = 1.7337 \pm 0.0004$ and is also unchanged. The exponent $\tau_1$ of the mean cluster length is related to $\beta_1$ and $\nu_{1\parallel}$ by the scaling relation $\nu_{1\parallel} = \beta_1 + \tau_1$ and using the above estimates yields $\tau_1 = 1$ to within the accuracy of our results. We conjecture that this value of $\tau_1$ is exact and further support for the conjecture is provided by the direct series expansion estimate $\tau_1= 1.0002 \pm 0.0003$.Comment: 12pages LaTeX, ioplppt.sty, to appear in J. Phys.

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=2d=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

Correction-to-scaling exponents for two-dimensional self-avoiding walks

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ~0.01% accuracy up to N = 4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Delta_1 = 3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.Comment: LaTeX 2.09, 56 pages. Version 2 adds a renormalization-group discussion near the end of Section 2.2, and makes many small improvements in the exposition. To be published in the Journal of Statistical Physic

Perimeter Generating Functions For The Mean-Squared Radius Of Gyration Of Convex Polygons

We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the generating functions. In all cases the size exponent $\nu=1$.Comment: 8 pages, 1 figur

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

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

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 $\beta$ 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: $q_c = 0.4043528 \pm 0.0000010$ and $\beta = 0.27645 \pm 0.00010$; in the bond case: $q_c = 0.52198\pm 0.00001$ and $\beta = 0.2769\pm 0.0010$; and in the site-bond case: $q_c = 0.264173 \pm 0.000003$ and $\beta = 0.2766 \pm 0.0003$. 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 $\Delta = 1$.Comment: 26 pages, LaTeX. To appear in J. Phys.

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