An upper bound on the number of self-avoiding polygons via joining

For d≥2d≥2 and n∈Nn∈N even, let pn=pn(d)pn=pn(d) denote the number of length nn self-avoiding polygons in ZdZd up to translation. The polygon cardinality grows exponentially, and the growth rate limn∈2Np1/nn∈(0,∞)limn∈2Npn1/n∈(0,∞) is called the connective constant and denoted by μμ. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that pnμ−n≤Cn−1/2pnμ−n≤Cn−1/2 in dimension d=2d=2. Here, we establish that pnμ−n≤n−3/2+o(1)pnμ−n≤n−3/2+o(1) for a set of even nn of full density when d=2d=2. We also consider a certain variant of self-avoiding walk and argue that, when d≥3d≥3, an upper bound of n−2+d−1+o(1)n−2+d−1+o(1) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.