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Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\\vec G(n,\\lceil\\alpha n\\rceil,v)$ as $n\\to\\infty$. We prove this conjecture.\n  Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto"},"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":"2606.10214","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T22:08:54Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"5cc9ae519c7256c21358d025933d03a19c5d3b616ce79104fd013c45ba2349a3","abstract_canon_sha256":"1f61317e78c0d21ef221e563de143896396061add510b7518cfa319554302a76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:08:59.892832Z","signature_b64":"xKpKO1T7oVYiB5UpF+6Uk9IjpPuBuaS6JyRT6Sho/JoI/AM1vr1tD5B4FzpZ5THa0/Hyhot22EJlZKglDm28Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b4bb9c7511dfead08eaff0538984deaac4919064b3aa591836bad959f59eee9","last_reissued_at":"2026-06-10T01:08:59.891893Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:08:59.891893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distribution of Sandpile groups of random directed bipartite graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Deepesh Singhal","submitted_at":"2026-06-08T22:08:54Z","abstract_excerpt":"Fix a prime $p$ and a constant $\\frac{1}{p}<\\alpha\\leq 1$. Consider the random directed Erd\\H{o}s--R\\'enyi bipartite graph $\\vec G(n,\\lceil\\alpha n\\rceil ,v)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\\lceil\\alpha n\\rceil$, and edge probability $0<v<1$. Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\\vec G(n,\\lceil\\alpha n\\rceil,v)$ as $n\\to\\infty$. We prove this conjecture.\n  Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10214","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/2606.10214/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.10214","created_at":"2026-06-10T01:08:59.892050+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.10214v1","created_at":"2026-06-10T01:08:59.892050+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10214","created_at":"2026-06-10T01:08:59.892050+00:00"},{"alias_kind":"pith_short_12","alias_value":"NNF3TR2RDX7K","created_at":"2026-06-10T01:08:59.892050+00:00"},{"alias_kind":"pith_short_16","alias_value":"NNF3TR2RDX7K2CHK","created_at":"2026-06-10T01:08:59.892050+00:00"},{"alias_kind":"pith_short_8","alias_value":"NNF3TR2R","created_at":"2026-06-10T01:08:59.892050+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K","json":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K.json","graph_json":"https://pith.science/api/pith-number/NNF3TR2RDX7K2CHK74CTRGCN5K/graph.json","events_json":"https://pith.science/api/pith-number/NNF3TR2RDX7K2CHK74CTRGCN5K/events.json","paper":"https://pith.science/paper/NNF3TR2R"},"agent_actions":{"view_html":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K","download_json":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K.json","view_paper":"https://pith.science/paper/NNF3TR2R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.10214&json=true","fetch_graph":"https://pith.science/api/pith-number/NNF3TR2RDX7K2CHK74CTRGCN5K/graph.json","fetch_events":"https://pith.science/api/pith-number/NNF3TR2RDX7K2CHK74CTRGCN5K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K/action/storage_attestation","attest_author":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K/action/author_attestation","sign_citation":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K/action/citation_signature","submit_replication":"https://pith.science/pith/NNF3TR2RDX7K2CHK74CTRGCN5K/action/replication_record"}},"created_at":"2026-06-10T01:08:59.892050+00:00","updated_at":"2026-06-10T01:08:59.892050+00:00"}