{"paper":{"title":"Self-avoiding walks crossing a square","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"cond-mat.stat-mech","authors_text":"A. J. Guttmann, I. Jensen, M. Bousquet-M\\'elou","submitted_at":"2005-06-14T23:25:50Z","abstract_excerpt":"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, $1.628 < \\lambda < 1.782.$ We give exact results for the number of SAW of length $2L + 2K$ for $K = 0, 1, 2$ and asymptotic results for $K = o(L^{1/3})$.\n  We also consider the model in which a weight or {\\em fugacity}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cond-mat/0506341","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}