{"paper":{"title":"Nonconcentration of hitting times for random walks on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Hitting times of random walks on graphs have variance at least the square of their mean divided by one plus the log of the number of vertices.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Rafael Chiclana","submitted_at":"2026-05-17T15:57:39Z","abstract_excerpt":"We study nonconcentration of hitting times for simple random walk on finite graphs. We prove that, for every connected graph with $n$ vertices, \\[ \\operatorname{Var}_x(\\tau_y)+\\mathbb E_x\\tau_y \\ge \\frac{(\\mathbb E_x\\tau_y)^2}{1+\\log n}, \\] with the logarithmic term sharp up to constants. Under a bounded-degree assumption the additive mean term can be removed, giving a variance lower bound depending only on \\(\\mathbb E_x\\tau_y\\) and the graph distance \\(\\dist(x,y)\\). We show that this degree assumption is necessary by constructing high-degree graphs with linear mean and bounded variance; the s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every connected graph with n vertices, Var_x(τ_y) + E_x τ_y ≥ (E_x τ_y)^2 / (1 + log n), with the logarithmic term sharp up to constants; under bounded degree the additive term can be removed.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The random walk is the simple symmetric random walk on an undirected connected finite graph; the constructions rely on specific high-degree vertex placements that keep variance bounded while making mean linear.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Establishes variance lower bounds for hitting times of random walks on graphs and disproves a conjecture on local nonconcentration via high-degree constructions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Hitting times of random walks on graphs have variance at least the square of their mean divided by one plus the log of the number of vertices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9ae38250d799ed2016c7db10847b4f266af5fa828564c963d2516ee97f19c5e7"},"source":{"id":"2605.17513","kind":"arxiv","version":1},"verdict":{"id":"2ccd9d4d-dc01-4a27-8a7d-3f6f448a8b71","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:37:18.247183Z","strongest_claim":"For every connected graph with n vertices, Var_x(τ_y) + E_x τ_y ≥ (E_x τ_y)^2 / (1 + log n), with the logarithmic term sharp up to constants; under bounded degree the additive term can be removed.","one_line_summary":"Establishes variance lower bounds for hitting times of random walks on graphs and disproves a conjecture on local nonconcentration via high-degree constructions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The random walk is the simple symmetric random walk on an undirected connected finite graph; the constructions rely on specific high-degree vertex placements that keep variance bounded while making mean linear.","pith_extraction_headline":"Hitting times of random walks on graphs have variance at least the square of their mean divided by one plus the log of the number of vertices."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17513/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.518429Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:51:32.003569Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.653064Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.629291Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"9be09ad45f3257a1042cbfa8eea89d9d6c67df878e8cdfb144fc68d31b86fe69"},"references":{"count":54,"sample":[{"doi":"10.1214/17-aop1189","year":2018,"title":"Random walks on the random graph , JOURNAL =","work_id":"96ff4b03-be4e-423b-bba5-b83a65193961","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1215/00127094-2010-029","year":2010,"title":"Lubetzky, Eyal and Sly, Allan , TITLE =. 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