Appendix: Series Data and Growth Constant, Amplitude and Exponent Estimates

In this appendix we have gathered together the series expansions for self-avoiding polygons on square, honeycomb and triangular lattices enumerated by either perimeter or area and the counts for the number of polyominoes on the same lattices. In addition we provide data for the number of SAP on three-dimensional lattices and the number of three-dimensional polyominoes (or polycubes).

Below we also provide a listing for the estimated growth constants¹, critical amplitudes and critical exponents for these problems. For any lattice, the growth constant for SAP and SAW is the same. The amplitude B for polygons is defined through p _n ~ Bμⁿ n ^α−3. For polyominoes if the growth constant is τ, the amplitude B is defined by assuming the number of n-celled polyominoes grows as Bτⁿ/n, while for polycubes the corresponding expression is Bτⁿ/n ^1.5. In estimating the amplitudes of polygons, we used the value of the growth constant μ in the table below and assumedα= 0.5 for two-dimensional lattices, andα= 0.23721 for three-dimensional lattices. The analysis of the amplitudes assumed only analytic correction-to-scaling terms. For the two-dimensional problems we believe this to be appropriate, while for the three-dimensional problems it is generally believed that there are non-analytic corrections to scaling, but the data we have is so limited that incorporating this refinement into the analysis is probably not justified.