{"paper":{"title":"Second-order fluctuations for a phase transition in random partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jaime Garza, Yizao Wang","submitted_at":"2026-07-02T09:38:01Z","abstract_excerpt":"In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameter $\\alpha\\in(0,1)$ and $\\theta>-\\alpha$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\\{1,\\ldots,n\\}$ and consider $j=j_n\\to\\infty$ as $n\\to\\infty$. They revealed a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\\sim rn^{\\alpha/(1+\\alpha)}$ for some $r>0$. A natural next question is to understand the corresponding second-order "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01946","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01946/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}