{
  "id": "1212.3822",
  "version": "v2",
  "published": "2012-12-16T19:37:15.000Z",
  "updated": "2013-09-30T02:02:42.000Z",
  "title": "The Satisfiability Threshold for $k$-XORSAT, using an alternative proof",
  "authors": [
    "Boris Pittel",
    "Gregory B. Sorkin"
  ],
  "comment": "This version integrates the previous version's alternative proof into the paper (see arXiv:1212.1905). Other proofs are amended, and the paper's structure is clarified. The main result is improved: the phase transition occurs for an arbitrarily slowly growing gap between m and n",
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider \"unconstrained\" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\\mathbb{F}_2$ over $n$ variables, each equation containing $k \\ge 3$ variables, and also consider a \"constrained\" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the threshold for unconstrained 3-XORSAT. We show that $m/n=1$ remains a sharp threshold for satisfiability of constrained $k$-XORSAT for every $k \\ge 3$, and we use standard results on the 2-core of a random $k$-uniform hypergraph to extend this result to find the threshold for unconstrained $k$-XORSAT. For constrained $k$-XORSAT we narrow the phase transition window, showing that $n-m \\to \\infty$ implies almost-sure satisfiability, while $m-n \\to \\infty$ implies almost-sure unsatisfiability.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-30T02:02:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "satisfiability threshold",
      "alternative proof",
      "sharp threshold",
      "implies almost-sure unsatisfiability",
      "phase transition window"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.3822P"
    }
  }
}