{"paper":{"title":"Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Elchanan Mossel, Gil Kalai","submitted_at":"2010-11-16T01:27:33Z","abstract_excerpt":"A key fact in the theory of Boolean functions $f : \\{0,1\\}^n \\to \\{0,1\\}$ is that they often undergo sharp thresholds. For example: if the function $f : \\{0,1\\}^n \\to \\{0,1\\}$ is monotone and symmetric under a transitive action with $\\E_p[f] = \\eps$ and $\\E_q[f] = 1-\\eps$ then $q-p \\to 0$ as $n \\to \\infty$. Here $\\E_p$ denotes the product probability measure on $\\{0,1\\}^n$ where each coordinate takes the value $1$ independently with probability $p$. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.3566","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":""},"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"}