Self-avoiding walks crossing a square

Abstract

We study a restricted class of self-avoiding walks (SAW) which start at the origin (0, 0), end at $(L,L)$, and are entirely contained in the square $[0,L]\times [0,L]$ on the square lattice ${\mathbb{Z}}^2$. The number of distinct walks is known to grow as ${\lambda }^{L^2+o(L^2)}$. We estimate $\lambda =1.744550\pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628<\lambda <\mathrm{1.782.}$ We give exact results for the number of SAW of length $2L+2K$ for $K=0,1,2$ and asymptotic results for $K=o(L^{1/3})$. We also consider the model in which a weight or {\em fugacity} $x$ is associated with each step of the walk. This gives rise to a canonical model of a phase transition. For $x<1/\mu$ the average length of a SAW grows as $L$, while for $x>1/\mu$ it grows as $L^2$. Here $\mu$ is the growth constant of unconstrained SAW in ${\mathbb{Z}}^2$. For $x=1/\mu$ we provide numerical evidence, but no proof, that the average walk length grows as $L^{4/3}$. We also consider Hamiltonian walks under the same restriction. They are known to grow as ${\tau }^{L^2+o(L^2)}$ on the same $L\times L$ lattice. We give precise estimates for $\tau$ as well as upper and lower bounds, and prove that $\tau <\lambda .$

27 pages, 9 figures. Paper updated and reorganised following refereeing

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