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- W2891076703 abstract "Let $c_n = c_n(d)$ denote the number of self-avoiding walks of length $n$ starting at the origin in the Euclidean nearest-neighbour lattice $mathbb{Z}^d$. Let $mu = lim_n c_n^{1/n}$ denote the connective constant of $mathbb{Z}^d$. In 1962, Hammersley and Welsh [HW62] proved that, for each $d geq 2$, there exists a constant $C > 0$ such that $c_n leq exp(C n^{1/2}) mu^n$ for all $n in mathbb{N}$. While it is anticipated that $c_n mu^{-n}$ has a power-law growth in $n$, the best known upper bound in dimension two has remained of the form $n^{1/2}$ inside the exponential. The natural first improvement to demand for a given planar lattice is a bound of the form $c_n leq exp (C n^{1/2 - epsilon})mu^n$, where $mu$ denotes the connective constant of the lattice in question. We derive a bound of this form for two such lattices, for an explicit choice of $epsilon > 0$ in each case. For the hexagonal lattice $mathbb{H}$, the bound is proved for all $n in mathbb{N}$; while for the Euclidean lattice $mathbb{Z}^2$, it is proved for a set of $n in mathbb{N}$ of limit supremum density equal to one. A power-law upper bound on $c_n mu^{-n}$ for $mathbb{H}$ is also proved, contingent on a non-quantitative assertion concerning this lattice's connective constant." @default.
- W2891076703 created "2018-09-27" @default.
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- W2891076703 date "2018-09-04" @default.
- W2891076703 modified "2023-09-27" @default.
- W2891076703 title "Bounding the number of self-avoiding walks: Hammersley-Welsh with polygon insertion" @default.
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