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Points at distance $d$ are joined with probability $\\min(1,\\beta/d)^2$, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for $\\lambda\\beta<1$ almost surely all connected components are finite, while for $\\lambda\\beta\\geq 31$ an infinite"},"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":"2608.12954","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-13T08:33:15Z","cross_cats_sorted":[],"title_canon_sha256":"6ca28ea6e06a7fbbaeb0080268963c1c10b0e7371b7f95a493afeb46c8c0e604","abstract_canon_sha256":"9e0b03eab7e0057b2b2f1dbc92178e44f0d2e373fd30333956947d5005c61ce5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-14T00:59:36.552339Z","signature_b64":"HFBqsb2KdF6G5OVZFQSf3UMcz8Vjza9XpFP5lOk5CXfcuF/v7XEnNSVYMRis8H9Ysm7LcqOc0VWQk6ZQgU64BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51c9c32c93019ec147fe5cd94a5c776f52fb84ffc4f786b190343ec9a45bc44e","last_reissued_at":"2026-08-14T00:59:36.546332Z","signature_status":"signed_v1","first_computed_at":"2026-08-14T00:59:36.546332Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rainbow percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Benjamin Lees, Peter Gracar","submitted_at":"2026-08-13T08:33:15Z","abstract_excerpt":"We consider the weight-dependent random connection model on a Poisson point process of intensity $\\lambda$ on $\\mathbb{R}\\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\\vee s)|x-y|\\leq\\beta$. 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