{"paper":{"title":"Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.PR","authors_text":"Emmanuel Abbe, Katherine Edwards","submitted_at":"2015-04-30T17:48:21Z","abstract_excerpt":"This paper shows that the logarithm of the number of solutions of a random planted $k$-SAT formula concentrates around a deterministic $n$-independent threshold. Specifically, if $F^*_{k}(\\alpha,n)$ is a random $k$-SAT formula on $n$ variables, with clause density $\\alpha$ and with a uniformly drawn planted solution, there exists a function $\\phi_k(\\cdot)$ such that, besides for some $\\alpha$ in a set of Lesbegue measure zero, we have $ \\frac{1}{n}\\log Z(F^*_{k}(\\alpha,n)) \\to \\phi_k(\\alpha)$ in probability, where $Z(F)$ is the number of solutions of the formula $F$. This settles a problem lef"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08316","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"}