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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}. 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