{"paper":{"title":"Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Leonid V. Bogachev","submitted_at":"2011-11-14T18:24:21Z","abstract_excerpt":"We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form $\\mathcal{F}(z)=\\prod_{\\ell=1}^\\infty \\mathcal{F}_0(z^\\ell)$ (which entails equal weighting among possible parts $\\ell\\in\\mathbb{N}$). Under mild technical assumptions on the function $H_0(u)=\\ln(\\mathcal{F}_0(u))$, we show that the limit shape $\\omega^*(x)$ exists and is given by the equation $y=\\gamma^{-1}H_0(\\mathrm{e}^{-\\gamma x})$, where $\\gamma^2=\\int_0^1 u^{-1}H_0(u)\\,\\mathrm{d}u$. The wide class "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3311","kind":"arxiv","version":4},"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"}