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The resistance between two nodes in general position on a fan network with n radial lines and m transverse lines is determined. Also a similar result of Izmailian, Kenna and Wu [7] for an $m\times n$ cobweb network is reproduced but the method used here is significantly different. It avoids the use of the Kirchhoff matrix, requires the solution of just one instead of two eigenvalue problems and results directly in only a single summation. Further the current distribution is given explicitly as a biproduct of the method. The method is the same as that used by Tan, Zhou and Yang [10] to find the cobweb resistance between center and perimeter for $1\le m\le3$ and general $n$. Proof of their conjecture for general $m$ is discussed., Comment: 13 pages, 4 figures

We consider the problem of two-point resistance on an m x n cobweb network with a superconducting boundary, which is topologically equivalent to a geographic globe. We deduce a concise formula for the resistance between any two nodes on the globe using a method of direct summation pioneered by one of us [Z. Z. Tan, et al, J. Phys. A 46, 195202 (2013)]. This method contrasts the Laplacian matrix approach which is difficult to apply to the geometry of a globe. Our analysis gives the result directly as a single summation., Comment: 15 pages, 3 figures

We investigate the behaviour of the mean size of directed compact percolation clusters near a damp wall in the low-density region, where sites in the bulk are wet (occupied) with probability $p$ while sites on the wall are wet with probability $p_w$. Methods used to find the exact solution for the dry case ($p_w=0$) and the wet case ($p_w=1$) turn out to be inadequate for the damp case. Instead we use a series expansion for the $p_w=2p$ case to obtain a second order inhomogeneous differential equation satisfied by the mean size, which exhibits a critical exponent $\gamma=2$, in common with the wet wall result. For the more general case of $p_w=rp$, with $r$ rational, we use a modular arithmetic method of finding ODEs and obtain a fourth order homogeneous ODE satisfied by the series. The ODE is expressed exactly in terms of $r$. We find that in the damp region $0

An m x n cobweb network consists of n radial lines emanating from a center and connected by $m$ concentric n-sided polygons. A conjecture of Tan, Zhou and Yang for the resistance from center to perimeter of the cobweb is proved by extending the method used by the above authors to derive formulae for m = 1, 2 and 3 and general n. The resistance of an m x (s+t+1) fan network from the apex to a point on the boundary distant s from the corner is also found., Comment: 7 pages, 3 figures

We study the corner-to-corner resistance of an M x N resistor network with resistors r and s in the two spatial directions, and obtain an asymptotic expansion of its exact expression for large M and N. For M = N, r = s =1, our result is R_{NxN} = (4/pi) log N + 0.077318 + 0.266070/N^2 - 0.534779/N^4 + O(1/N^6)., Comment: 12 pages, re-arranged sections

We state and prove several theorems that demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices of a generalised inhomogeneous five-vertex model on the square lattice, given certain conditions hold, is equivalent to the Gessel-Viennot determinant for the number of configurations of $N$ non-intersecting directed lattice paths, or vicious walkers, with various boundary conditions. Our theorems are sufficiently general to allow generalisation to any regular planar lattice.

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. A