The Hammersley-Welsh bound for self-avoiding walk revisited

2017/08/30 by Hutchcroft, Tom
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1708.09460

Abstract

The Hammersley-Welsh bound (1962) states that the number c_n of length n self-avoiding walks on ℤ^d satisfies c_n ≤ exp [ O(n^1/2) ] μ_cⁿ, where μ_c=μ_c(d) is the connective constant of ℤ^d. While stronger estimates have subsequently been proven for d≥ 3, for d=2 this has remained the best rigorous, unconditional bound available. In this note, we give a new, simplified proof of this bound, which does not rely on the combinatorial analysis of unfolding. We also prove a small, non-quantitative improvement to the bound, namely c_n ≤ exp[ o(n^1/2)] μ_cⁿ. The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.

Related

• Upper bounds for the connective constant of weighted self-avoiding walks
• New upper bound for the connective constant for square-lattice self-avoiding walks
• Lectures on Self-Avoiding Walks
• Self-attracting self-avoiding walk
• Self-avoiding Walks of Lattice Strips