Extremal cuts of sparse random graphs

Abstract

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 $n(\frac{\gamma}{4}+\mathsf{P}_{*}\sqrt{\frac{\gamma}{4}}+o(\sqrt{\gamma}))+o(n)$ while the size of the minimum bisection is $n(\frac{\gamma}{4}-\mathsf{P}_{*}\sqrt{\frac{\gamma}{4}}+o(\sqrt{\gamma}))+o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington–Kirkpatrick model, with $\mathsf{P}_{*}\approx0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi’s formula.