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

For d.2 and n.N even, let pn=pn(d) denote the number of length n self-avoiding polygons in Zd up to translation. The polygon cardinality grows exponentially, and the growth rate limn.2Np1/nn.(0,.) is called the connective constant and denoted by .. Madras [J. Stat. Phys. 78 (1995) 681.699] has shown that pn..n.Cn.1/2 in dimension d=2. Here, we establish that pn..n.n.3/2+o(1) for a set of even n of full density when d=2. We also consider a certain variant of self-avoiding walk and argue that, when d.3, an upper bound of n.2+d.1+o(1) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.