{"paper":{"title":"Phase Transitions in Turnpike Theory For Mean-Field Games","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Siddharth Karuturi","submitted_at":"2026-04-24T23:10:34Z","abstract_excerpt":"We study a translation-invariant mean-field game on the flat torus with interaction $F(x,m)=\\gamma (K*m)(x)$, where $K$ is smooth, even, and mean-zero. The interaction is of potential type, arising as the first variation of a quadratic energy, though the stationary system is not treated variationally.\n  Linearizing around the uniform equilibrium yields mode-wise $2\\times 2$ systems with dispersion $\\sigma_\\xi(\\gamma)=\\nu^2(2\\pi|\\xi|)^4+\\gamma(2\\pi|\\xi|)^2\\hat K(\\xi)$. If $\\hat K$ is negative for some mode, a finite threshold \\[ \\gamma_c=\\min_{\\hat K(\\xi)<0}\\frac{\\nu^2(2\\pi|\\xi|)^2}{|\\hat K(\\xi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20213","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/2605.20213/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"}