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

2018/08/29 by Hammond, Alan
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1808.09597

Abstract

For d ≥ 2 and n ∈ ℕ, let W_n denote the uniform law on self-avoiding walks of length n beginning at the origin in the nearest-neighbour integer lattice ℤ^d, and write Γ for a W_n-distributed walk. We show that the closing probability W_n ( \vert\vert Γ_n \vert\vert = 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.

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