{"paper":{"title":"Inside the clustering threshold for random linear equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"cs.DM","authors_text":"Michael Molloy, Pu Gao","submitted_at":"2013-09-19T15:01:43Z","abstract_excerpt":"We study a random system of $cn$ linear equations over $n$ variables in GF(2), where each equation contains exactly $r$ variables; this is equivalent to $r$-XORSAT. \\cite{ikkm,amxor} determined the clustering threshold, $c^*_r$: if $c=c^*_r+\\e$ for any constant $\\e>0$, then \\aas the solutions partition into well-connected, well-separated {\\em clusters} (with probability tending to 1 as $n\\rightarrow\\infty$). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but most"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6651","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"}