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The satisfiability threshold and solution geometry of random linear equations over finite fields Jane Gao (jane.gao@monash.edu) School of Mathematical Sciences Examples k-SAT k-NAESAT k-XORSAT k-colouring of graphs

  1. The satisfiability threshold and solution geometry of random linear equations over finite fields Jane Gao (jane.gao@monash.edu) School of Mathematical Sciences

  2. Examples • k-SAT • k-NAESAT • k-XORSAT • k-colouring of graphs • 2-colouring of k-uniform hyper graphs • Potts model • Ising model (ferromagnetic, antiferromagnetic) • graph matching (monomer-dimer model) • independent set (hard core model) • SK model • mixed p-spin model ‹#›

  3. ‹#›

  4. K-SAT threshold ‹#›

  5. Main challenge: large deviation ‹#›

  6. Phase transitions of the solution geometry [Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova 2007] ‹#›

  7. Lottery effect kills the second moment method ‹#›

  8. ‹#›

  9. Replica symmetry (correlation decay) ‹#›

  10. Solution clusters ‹#›

  11. Bethe free entropy ‹#›

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