Self-avoiding polygons on the square lattice

Abstract

We have developed an improved algorithm that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 90. Analysis of the resulting series yields very accurate estimates of the connective constant µ = 2.63815852927(1) (biased) and the critical exponent α = 0.5000005(10) (unbiased). The critical point is indistinguishable from a root of the polynomial 581x 4 + 7x 2 − 13 = 0. An asymptotic expansion for the coefficients is given for all n. There is strong evidence for the absence of any non-analytic correction-to-scaling exponent.

Punctured polygons and polyominoes on the square lattice

“…A more accurate estimate of the growth constant for SAP enumerated by area has been given, complementing our earlier work on the perimeter growth constants [17].…”

“…Note that these error bounds should not be viewed as a measure of the true error as they cannot include possible systematic sources of error. However systematic error can also be taken into account in favourable situations, as for example in the case of SAP enumerated by perimeter [17]. Again, in the interests of space, we present only our results, and not the intermediate detail from which our estimates were made.…”

“…For unpunctured SAP, the perimeter generating function was recently extended [17] to 90 step polygons, and the asymptotics clearly identified. The polygon generating function is defined to be…”

“…The most recent result for polygon perimeters appears to be [17] where polygons of perimeter up to 90 steps are given. In that paper our analysis of the polygon perimeter generating function led us to conclude that…”

“…Given such a series, using the numerical technique known as differential approximants [13], highly accurate estimates can frequently be obtained for the critical point and exponents, as well as the location and critical exponents of possible non-physical singularities. Other numerical methods are discussed in [17], and those used in the present study are described more fully below.…”