## 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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