{"paper":{"title":"Diffuse Interface Energies with Microscopic Heterogeneities II: Rare Events","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Christian Wagner, Peter S. Morfe","submitted_at":"2026-06-16T14:21:29Z","abstract_excerpt":"We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\\delta$ of the heterogeneities is much smaller (microscopic) than the interface width $\\epsilon$ (mesoscopic). In a companion paper, we show that if the ratio $\\epsilon^{-1} \\delta$ vanishes fast enough as $\\epsilon \\to 0$, then the functionals converge to an effective surface energy where the energy density is determined by homogenization effects originating at microscopic scales. Here we prove that if the ratio $\\epsilon^{-1} \\delta $ vanishes too slowly, the limit of the functional m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17968","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/2606.17968/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"}