The satisfiability threshold and solution geometry of random linear - PowerPoint PPT Presentation

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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

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 • 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 ‹#›
‹#›
K-SAT threshold ‹#›
Main challenge: large deviation ‹#›
Phase transitions of the solution geometry [Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova 2007] ‹#›
Lottery effect kills the second moment method ‹#›
‹#›
Replica symmetry (correlation decay) ‹#›
Solution clusters ‹#›
Bethe free entropy ‹#›

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