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Extremal cuts of sparse random graphs. (English) Zbl 1372.05196

Summary: For Erdős-Rényi random graphs with average degree \(\gamma\), and uniformly random \(\gamma\)-regular graph on \(n\) vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are \(n(\frac{\gamma}{4}+\mathsf{P}_\ast\sqrt{\frac{\gamma}{4}}+o(\sqrt{\gamma}))+o(n)\) while the size of the minimum bisection is \(n(\frac{\gamma}{4}-\mathsf{P}_\ast\sqrt{\frac{\gamma}{4}}+o(\sqrt{\gamma}))+o(n)\). Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with \(\mathsf{P}_\ast\approx0.7632\) standing for the ground state energy of the latter, expressed analytically via Parisi’s formula.

MSC:

05C80 Random graphs (graph-theoretic aspects)
05C42 Density (toughness, etc.)
68R10 Graph theory (including graph drawing) in computer science
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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