  For d at least two and integer n, let c_n = c_n(d) denote the number of
length n self-avoiding walks beginning at the origin in the integer lattice
Z^d, and, for even n, let p_n = p_n(d) denote the number of length n
self-avoiding polygons in Z^d up to translation. Then the probability under the
uniform law W_n on self-avoiding walks Gamma of any given odd length n
beginning at the origin that Gamma closes -- i.e., that Gamma's endpoint is a
neighbour 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 exponential
growth: lim_n c_n^{1/n} = lim_{n even} p_n^{1/n} = mu (where the common value
mu is called the connective constant). Madras [26] has shown that p_n is at
most C n^{-1/2} mu^n in dimension d=2, while the closing probability was
recently 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)} for
a subsequence of odd n, if d = 2; and (3) p_n is at most n^{-3/2 + o(1)} mu^n
for 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 of
self-avoiding walk.
