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If (i) $w \\gg N^{-2/d}$ then the limit is a PDE on ${\\bf R}^d$. If (ii) $N^{-2/d} \\gg w \\gg N^{-1}$, then the limit is an ODE. If (iii) $w \\ll N^{-1}$ then the effect of selection vanishes in the limit. In regime (ii) if we introduce a mutation $\\mu$ so that $\\mu /w \\to \\infty$ s"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1511.04713","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-15T15:30:07Z","cross_cats_sorted":[],"title_canon_sha256":"a4fb3078bdbd28c92f8940dbdb559ffa1e698c6ff7683eb9f7ceeccab93dbb75","abstract_canon_sha256":"b6c012e3963652fe36542b97ae1d41c5617d1112d6bf0cd41f74315e708c3c80"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:51.928230Z","signature_b64":"HkGb/zAMV3UufSLG/NIU6bPaba61B8qZIWFeVPQ61GigneQgqF3sJSro2/elCK41b4HclIuTo1aw8Rm+8d3oCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d66028ebf2ef8697bf51b184cca4bdd4f9e7bec6dfcc113e2ba4378fc3d4651","last_reissued_at":"2026-05-18T01:26:51.927526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:51.927526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Evolutionary Games on the Torus with Weak Selection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"J.T. Cox, Rick Durrett","submitted_at":"2015-11-15T15:30:07Z","abstract_excerpt":"We study evolutionary games on the torus with $N$ points in dimensions $d\\ge 3$. The matrices have the form $\\bar G = {\\bf 1} + w G$, where ${\\bf 1}$ is a matrix that consists of all 1's, and $w$ is small. As in Cox Durrett and Perkins \\cite{CDP} we rescale time and space and take a limit as $N\\to\\infty$ and $w\\to 0$. If (i) $w \\gg N^{-2/d}$ then the limit is a PDE on ${\\bf R}^d$. If (ii) $N^{-2/d} \\gg w \\gg N^{-1}$, then the limit is an ODE. If (iii) $w \\ll N^{-1}$ then the effect of selection vanishes in the limit. 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