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We prove that the smallest gaps between particles are typically of order $n^{-3/4}$, and that the associated joint point process of gap locations and gap sizes, after rescaling the gaps by $n^{3/4}$, converges to a Poisson point process. As a consequence, we show that the $k$-th smallest rescaled gap has a limiting density proportional to $x^{4k-1}e^{-\\frac{\\mathcal{J}}{4}x^{4}}$, where $\\mathcal{J}=\\pi^{2}\\int \\rho(z)^{3}d^{2}z$ and"},"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":"2507.23502","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-31T12:41:33Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"4502f6e9cf0b35f316b58703aa4243d3011efe6ad82467340e51e7084651bcd5","abstract_canon_sha256":"1b39ccd313afb391a02b81fa2e0ec70c91edf912ae1b461925c2e55ce43a8605"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:58:33.892102Z","signature_b64":"2r7PDL2ejcbLqGNmGqJwWgWtgZLUabjZU9u5fpjfWlhDl7vn9as5ExAR0utEt+PM0U8Pt+nFgM1VYxswoEULDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e473b08244b6609565a03e4e3e7b73b355fec105eaad504d07f0de8db5f2d2a","last_reissued_at":"2026-07-05T11:58:33.891615Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:58:33.891615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smallest gaps of the two-dimensional Coulomb gas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Christophe Charlier","submitted_at":"2025-07-31T12:41:33Z","abstract_excerpt":"We consider the two-dimensional Coulomb gas with a general potential at the determinantal temperature, or equivalently, the eigenvalues of random normal matrices. 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