{"paper":{"title":"Inverse problem of the limit shape for convex lattice polygonal lines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Leonid V. Bogachev, Sakhavat M. Zarbaliev","submitted_at":"2011-10-30T19:11:16Z","abstract_excerpt":"It is known that random convex polygonal lines on $\\mathbb{Z}_+^2$ (with the endpoints fixed at $0=(0,0)$ and $n=(n_1,n_2)\\to\\infty$) have a limit shape with respect to the uniform probability measure, identified as the parabola arc $\\sqrt{c\\myp(1-x_1)}+\\sqrt{x_2}=\\sqrt{c}$, where $n_2/n_1\\to c$. The present paper is concerned with the inverse problem of the limit shape. We show that for any strictly convex, $C^3$-smooth arc $\\gamma\\subset\\mathbb{R}_+^2$ starting at the origin, there is a probability measure $P_n^\\gamma$ on convex polygonal lines, under which the curve $\\gamma$ is their limit "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6636","kind":"arxiv","version":1},"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"}