Self-avoiding walks crossing a square

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 $λ^{L^2+o(L^2)}$. We estimate $λ= 1.744550 \pm 0.000005$ as well as obtaining strict upper and lower bounds, $1.628 < λ< 1.782.$ 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 $x < 1/μ$ the average length of a SAW grows as $L$, while for $x > 1/μ$ it grows as $L^2$. Here $μ$ is the growth constant of unconstrained SAW in ${\mathbb Z}^2$. For $x = 1/μ$ 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 $τ^{L^2+o(L^2)}$ on the same $L \times L$ lattice. We give precise estimates for $τ$ as well as upper and lower bounds, and prove that $τ< λ.$

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