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For d at least two and integer n, let c_n = c_n(d) denote the number oflength n self-avoiding walks beginning at the origin in the integer latticeZ^d, and, for even n, let p_n = p_n(d) denote the number of length nself-avoiding polygons in Z^d up to translation. Then the probability under theuniform law W_n on self-avoiding walks Gamma of any given odd length nbeginning at the origin that Gamma closes -- i.e., that Gamma's endpoint is aneighbour of the origin -- is given by W_n ( Gamma closes ) = 2(n+1)p_{n+1}/c_n. The polygon and walk cardinalities share a common exponentialgrowth: lim_n c_n^{1/n} = lim_{n even} p_n^{1/n} = mu (where the common valuemu is called the connective constant). Madras [26] has shown that p_n is atmost C n^{-1/2} mu^n in dimension d=2, while the closing probability wasrecently shown in [12] to satisfy W_n ( Gamma closes ) is at most n^{-1/4 +o(1)} in any dimension d at least two.  Here we establish that (1) W_n ( Gamma closes ) is at most n^{-1/2 + o(1)}for any d at least two; (2) W_n ( Gamma closes ) is at most n^{-4/7 + o(1)} fora subsequence of odd n, if d = 2; and (3) p_n is at most n^{-3/2 + o(1)} mu^nfor a set of even n of full density when d=2.  We also argue that the closing probability is bounded above by n^{-(1 - 1/d)+ o(1)} on a full density set when d is at least three for a certain variant ofself-avoiding walk.

On self-avoiding polygons and walks: counting, joining and closing