On self-avoiding polygons and walks: the snake method via polygon joining

For d >= 2 and n is an element of IN, let W-n, denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice Z(d), and write Gamma for a W-n,-distributed walk. We show that the closing probability W-n(parallel to Gamma(n)parallel to = 1) that Gamma's endpoint neighbours the origin is at most n(-4/7+o (1)) for a positive density set of odd n in dimension d = 2. This result is proved using the snake method, a general technique for proving closing probability upper bounds, which originated in [3] and was made explicit in [8]. Our conclusion is reached by applying the snake method in unison with a polygon joining technique whose use was initiated by Madras in [13].