The Annals of Probability

Extremal cuts of sparse random graphs

Amir Dembo, Andrea Montanari, and Subhabrata Sen

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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.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 1190-1217.

Received: March 2015
Revised: August 2015
First available in Project Euclid: 31 March 2017

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Max-cut bisection Erdős–Rényi graph regular graph Ising model spin glass Parisi formula


Dembo, Amir; Montanari, Andrea; Sen, Subhabrata. Extremal cuts of sparse random graphs. Ann. Probab. 45 (2017), no. 2, 1190--1217. doi:10.1214/15-AOP1084.

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