On self-avoiding polygons and walks: The snake method via pattern fluctuation

Abstract

For d ≥ 2 and n ∈ N, let Wn denote the uniform law on selfavoiding walks of length n beginning at the origin in the nearest-neighbour integer lattice Z d , and write Γ for a Wn-distributed walk. We show that the closing probability Wn ||Γn|| = 1 that Γ's endpoint neighbours the origin is at most n −1/2+o(1) in any dimension d ≥ 2. The method of proof is a reworking of that in [4], which found a closing probability upper bound of n −1/4+o(1) . A key element of the proof is made explicit and called the snake method. It is applied to prove the n −1/2+o(1) upper bound by means a technique of Gaussian pattern fluctuation.

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

“…This bound complements the n −1/4+o (1) general dimensional upper bound from [3]. In fact, the method of [3] was reworked in [8] to achieve the following n −1/2+o (1) bound.…”

“…We show that the closing probability Wn ||Γn|| = 1 that Γ'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].…”

“…In this section, we recall from [8] the snake method. The method is used to prove upper bounds on the closing probability, and assumes to the contrary that to some degree this probability has slow decay.…”

“…We have mentioned that the snake method is a probabilistic tool for proving closing probability upper bounds. It is introduced in [8]; (it originated, though was not made explicit, in [3]). The method states a condition that must be verified in order to achieve some such upper bound.…”

“…It is worth pointing out that their proofs are respectively almost trivial and fairly short (three to four pages in the latter case). It is thus suggested to the reader who wishes to make a moderate attempt at understanding the paper's inputs to read the proofs of these results from Sections 2.2 and 3.2 of [8] when their statements are reached. earlier version of [7].…”