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

For d >= 2 and n is an element of N even, let p(n) = p(n)(d) denote the number of length n self-avoiding polygons in Z(d) up to translation. The polygon cardinality grows exponentially, and the growth rate lim(n is an element of 2N) (1/n)(pn) is an element of (0,infinity) is called the connective constant and denoted by mu. Madras [J. Stat. Phys. 78 (1995) 681-699] has shown that p(n)mu(-n) <= Cn(-1/2) in dimension d = 2. Here, we establish that p(n)mu(-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.