Journal article

Self-avoiding walks crossing a square

M Bousquet-Melou, AJ Guttmann, I Jensen

JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL | IOP PUBLISHING LTD | Published : 2005

Abstract

We study a restricted class of self-avoiding walks (SAWs) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0, L] × [0, L] on the square lattice . The number of distinct walks is known to grow as . We estimate λ ≤ 1.744 550 ± 0.000 005 as well as obtaining strict upper and lower bounds, 1.628 < λ < 1.782. We give exact results for the number of SAWs of length 2L + 2K for K ≤ 0, 1, 2 and asymptotic results for K ≤ o(L1/3). We also consider the model in which a weight or 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 it grows as L 2. Here μ is the growth constant of unconstrain..

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