Extremal cuts of sparse random graphs

• Amir Dembo ^[1] ; Andrea Montanari ^[1] ; Subhabrata Sen ^[1]
  1. [1] Stanford University

     Stanford University

     Estados Unidos

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 2, 2017, págs. 1190-1217
• Idioma: inglés
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• Resumen

  • For Erdős–Rényi random graphs with average degree γγ, and uniformly random γγ-regular graph on nn vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are n(γ4+P∗γ4−−√+o(γ−−√))+o(n)n(γ4+P∗γ4+o(γ))+o(n) while the size of the minimum bisection is n(γ4−P∗γ4−−√+o(γ−−√))+o(n)n(γ4−P∗γ4+o(γ))+o(n). Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington–Kirkpatrick model, with P∗≈0.7632P∗≈0.7632 standing for the ground state energy of the latter, expressed analytically via Parisi’s formula.