Enumeration of self avoiding trails on a square lattice using a transfer matrix technique

We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding walk universality class. Assuming that trails behave like $A \lambda ^n n^{11 \over 32}$, we find $\lambda = 2.72062 \pm 0.000006$ and $A = 1.272 \pm 0.002$.Comment: To be published in J. Phys. A:Math Gen. Pages: 16 Format: RevTe

Low-Temperature Series for the Correlation Length in d=3d=3 Ising Model

We extend low-temperature series for the second moment of the correlation function in $d=3$ simple-cubic Ising model from $u^{15}$ to $u^{26}$ using finite-lattice method, and combining with the series for the susceptibility we obtain the low-temperature series for the second-moment correlation length to $u^{23}$. An analysis of the obtained series by inhomogeneous differential approximants gives critical exponents $2\nu^{\prime} + \gamma^{\prime} \approx 2.55$ and $2\nu^{\prime} \approx 1.27$.Comment: 13 pages + 5 uuencoded epsf figures in Latex, OPCT-94-

Trade Openness: An Australian Perspective

Australia’s external trade is relatively low compared with the size of its economy. Indeed, Australia’s openness ratio (exports plus imports as a proportion of GDP) in 2002 was the third-lowest among the 30 OECD countries. This paper seeks to understand Australia’s low openness by analysing the empirical determinants of aggregate country trade. We begin by estimating a standard gravity model of bilateral trade. Although the model appears to fit the bilateral data very well, it does a relatively poor job at fitting countries’ aggregate trade levels, with different methodologies sometimes providing highly conflicting results. The focus of the paper is an equation for country openness. Our equation explains a substantial amount of the variation in how much countries trade using a small number of explanatory variables. We find that the most important determinants of openness are population and a measure of distance to potential trade partners. Countries with larger populations trade less, as do countries that are relatively more remote. Furthermore, after controlling for trade policy there is little evidence of a positive correlation between openness and economic development. While gravity models suggest Australia trades much more than expected, the openness equation suggests that its level of trade is relatively close to what would be expected. The most important factors in explaining Australia’s low openness ratio are its distance to the rest of the world, and to a lesser extent its large geographic size.trade; outward orientation; economic geography; trade liberalisation

Counting Planar Eulerian Orientations

Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $const\cdot (1 - \mu x)/\log(1 - \mu x),$ where we conjecture that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices.Comment: 26 pages, 20 figure

Accurate Estimates of 3D Ising Critical Exponents Using the Coherent-Anomaly Method

An analysis of the critical behavior of the three-dimensional Ising model using the coherent-anomaly method (CAM) is presented. Various sources of errors in CAM estimates of critical exponents are discussed, and an improved scheme for the CAM data analysis is tested. Using a set of mean-field type approximations based on the variational series expansion approach, accuracy comparable to the most precise conventional methods has been achieved. Our results for the critical exponents are given by \alpha=\afin, \beta=\bfin, \gamma=\gfin and \delta=\dfin.Comment: 16 pages, latex, 1 postscript figur