{"paper":{"title":"Phase transitions for the noisy transformer model in arbitrary dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR","stat.ML"],"primary_cat":"math.AP","authors_text":"Kyunghoo Mun, Matthew Rosenzweig","submitted_at":"2026-06-03T17:49:43Z","abstract_excerpt":"We study the McKean--Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\\ge2$. There is a unique $\\beta_*^{(d)}>0$ such that \\begin{equation*}\n  \\frac{I_{d/2+1}(\\beta_*^{(d)})}{I_{d/2}(\\beta_*^{(d)})}=\\frac1d, \\end{equation*} where $I_\\nu$ is the modified Bessel function of the first kind. For $0<\\beta\\le \\beta_*^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \\begin{equation*}\n  K_\\#^{(d)}(\\beta)=\\f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05140","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/2606.05140/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"}