A R Conway

Algebraic techniques for enumerating self-avoiding walks on the square lattice

The authors describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3N/4 times a polynomial in N, and so the approach is greatly superior to direct counting techniques. They have enumerated walks of up to 39 steps. As a consequence, they are able to accurately estimate the critical point, critical exponent, and critical amplitude.

On two-dimensional percolation

We present new series data for both high- and low-density bond and site percolation on the square lattice. The series have been obtained by the finite-lattice method, and in all cases extend pre-existing series. An analysis of these series gives refined estimates of critical points, critical exponents and amplitudes for bond and site animals, and for the percolation probability and mean-size exponents.

Directed animals on two-dimensional lattices

Previously, directed animals on square and triangular lattices have been enumerated by area, and have been found to have simple generating functions, whilst the hexagonal lattice generating function has not been obtained. Directed animals on several new lattices are enumerated, one class of which is solved exactly. Directed animals by bonds (with and without loops) are also enumerated. In each case an asymptotic growth like n-1/2 mu n is observed and precise estimates for mu are given.

The number of three-choice polygons

A polynomial time enumeration method for the three-choice polygon model in two dimensions is given together with numerical analysis of the enumerated series and an argument supporting the asymptotic behaviour of the number of imperfect staircase polygons.

Enumerating 2D percolation series by the finite-lattice method: theory

This paper describes a transfer-matrix algorithm for enumeration of series of interest in percolation on the square lattice. It allows efficient generation of both low-temperature and high-temperature expansions, as well as the combinatorially interesting enumeration of undirected animals by area or perimeter, with moments of the other property.

Lower bound on the connective constant for square lattice self-avoiding walks

By enumerating irreducible bridges exactly up to 40 steps, and by obtaining a lower bound to the number of bridges with less than 125 steps, a lower bound for the connective constant for square lattice self-avoiding walks of 2.62 is obtained.

Three-dimensional terminally attached self-avoiding walks and bridges

We study terminally attached self-avoiding walks and bridges on the simple cubic lattice, both by series analysis and Monte Carlo methods. We provide strong numerical evidence supporting a scaling relation between self-avoiding walks, bridges, and terminally attached self-avoiding walks, and posit that a corresponding amplitude ratio is a universal quantity.

SOME EXACT RESULTS FOR MOMENTS OF 2D DIRECTED ANIMALS

Using computer enumerations and the algebraic approximant method of series analysis, several new exact results have been found for moments of width, perimeter and loops for directed animals on the square and triangular lattices. A proof using q-series is given for exact solutions for widths on both the square and triangular lattices.

Longitudinal size exponent for square-lattice directed animals

Series expansions for the longitudinal gyration radius of square-lattice directed-site animals to order N=39 are given. This doubles the length of the available series. Analysis of the series yields the estimate nu /sub ///=0.81722+or-0.00005, which excludes the conjecture nu /sub ///=9/11 based on phenomenological renormalization and shorter series, but is consistent with the later phenomenological renormalization estimate of Dhar: nu /sub ///=0.81733+or-0.00005.

An analysis of New South Wales electronic vote counting

We re-examine the 2012 local government elections in New South Wales, Australia. The count was conducted electronically using a randomised form of the Single Transferable Vote (STV). It was already well known that randomness does make a difference to outcomes in some seats. We describe how the process could be amended to include a demonstration that the randomness was chosen fairly. Second, and more significantly, we found an error in the official counting software, which caused a mistake in the count in the council of Griffith, where candidate Rina Mercuri narrowly missed out on a seat. We believe the software error incorrectly decreased Mercuris winning probability to about 10%---according to our count she should have won with 91% probability. The NSW Electoral Commission (NSWEC) corrected their code when we pointed out the error, and made their own announcement. We have since investigated the 2016 local government election (held after correcting the error above) and found two new errors. We notified the NSWEC about these errors a few days after they posted the results.