{"paper":{"title":"Scaling limit and density conjecture for activated random walk on the complete graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The stationary number of sleeping particles in activated random walk on the complete graph has a Gumbel scaling limit for sink probabilities in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Harley Kaufman, Josh Meisel, Matthew Junge","submitted_at":"2026-04-06T15:13:24Z","abstract_excerpt":"We study driven-dissipative activated random walk with sleep probability $p$ on an $n$-vertex complete graph with a sink that traps jumping particles with probability $q_n$. We show that the number of sleeping particles $S_n$ left by the stationary distribution has a Gumbel scaling limit for $\\exp(-n^{1/3}) \\ll q_n \\ll n^{-1/2}$. The particular scaling implies that $S_n$ is hyperuniform and thus the stationary configuration law has negative correlations and is not a product measure. We also prove that $S_n/n$ converges to $p$ if and only if $q_n = e^{-o(n)}$, and that, when $q_n=0$, the number"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the number of sleeping particles S_n left by the stationary distribution has a Gumbel scaling limit for exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}. This implies that the stationary configuration law is not a product measure. We also prove that S_n/n converges to p if and only if q_n = e^{-o(n)}, and that, when q_n=0, the number of jumps to stabilization undergoes a phase transition at density p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proofs rely on the complete-graph mean-field structure and the precise asymptotic window for q_n; if the graph were not complete or the window violated, the Gumbel limit and the iff convergence statement would not necessarily hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Activated random walk on the complete graph with sink has Gumbel scaling for sleeping particles when exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}, density converges to p only for exponentially weak sinks, and jumps to stabilization phase-transition at density p when q_n=0.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The stationary number of sleeping particles in activated random walk on the complete graph has a Gumbel scaling limit for sink probabilities in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"18b6bf056595b4bc693930cfaf9b2479ede7bb928cbf1ca8c1ab0e97dae062d9"},"source":{"id":"2604.04747","kind":"arxiv","version":2},"verdict":{"id":"3e0d3e4e-081b-4bc2-bc95-a2d85a5bbc70","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T19:57:55.501881Z","strongest_claim":"We show that the number of sleeping particles S_n left by the stationary distribution has a Gumbel scaling limit for exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}. This implies that the stationary configuration law is not a product measure. We also prove that S_n/n converges to p if and only if q_n = e^{-o(n)}, and that, when q_n=0, the number of jumps to stabilization undergoes a phase transition at density p.","one_line_summary":"Activated random walk on the complete graph with sink has Gumbel scaling for sleeping particles when exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}, density converges to p only for exponentially weak sinks, and jumps to stabilization phase-transition at density p when q_n=0.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The proofs rely on the complete-graph mean-field structure and the precise asymptotic window for q_n; if the graph were not complete or the window violated, the Gumbel limit and the iff convergence statement would not necessarily hold.","pith_extraction_headline":"The stationary number of sleeping particles in activated random walk on the complete graph has a Gumbel scaling limit for sink probabilities in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.04747/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"}