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

Abstract

For d ≥ 2 and n ∈ N, let Wn denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice Z d , and write Γ for a Wndistributed walk. 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].

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

“…I am very grateful to a referee for a thorough discussion of the article [13]. Indeed, the present form of the two main theorems in the present article is possible on the basis of a suggestion made by this referee, (and this strengthened form has led to an improvement in Theorem 1.9(1), proved in [15] 2.0.3. Notation for certain corners of polygons.…”

“…It may be apparent that this paper shares with [14] and [15] certain combinatorial and probabilistic elements. The present article, and the other two, may be read alone, but there may be also be value in viewing the results in unison.…”

“…The snake method is applied via a technique of Gaussian pattern fluctuation in [14] to obtain Theorem 1.8. A second application of the snake method is made in [15] in order to reach a stronger inference regarding the closing probability, valid when d = 2, for a subsequence of even n. In this case, the snake method is allied not with a pattern fluctuation technique but instead with the tool that is central to the proofs of the present article's principal results Theorems 1.3 and 1.5: the polygon joining technique, initiated by Madras in [21]. The conclusion that is reached in [15] is now stated.…”

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

“…It is this first step that will be carried out in Section 4, by means of a technique of Gaussian pattern fluctuation. (In [10], this step is completed via polygon joining to prove Theorem 1.3. )…”

“…The snake method via polygon joining. In [10], the second application of the snake method appears. The sufficient condition in the method is not verified by pattern fluctuation but instead by a technique of polygon joining.…”