Communications in Mathematical Physics

, Volume 341, Issue 2, pp 435–489 | Cite as

Satisfiability Threshold for Random Regular nae-sat

  • Jian Ding
  • Allan Sly
  • Nike Sun


We consider the random regular k-nae- sat problem with n variables, each appearing in exactly d clauses. For all k exceeding an absolute constant \({{\it k}_0}\), we establish explicitly the satisfiability threshold \({{{d_\star} \equiv {d_\star(k)}}}\). We prove that for \({{d < d_\star}}\) the problem is satisfiable with high probability, while for \({{d > d_\star}}\) the problem is unsatisfiable with high probability. If the threshold \({{d_\star}}\) lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krz̧akała et al. [Proc Natl Acad Sci 104(25):10318–10323, 2007]. Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.


Partition Function Free Variable Gibbs Measure Empirical Measure Incoming Message 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ChicagoChicagoUSA
  2. 2.Department of StatisticsUniversity of California, BerkeleyBerkeleyUSA
  3. 3.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  4. 4.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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