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March 2017 Extremal cuts of sparse random graphs
Amir Dembo, Andrea Montanari, Subhabrata Sen
Ann. Probab. 45(2): 1190-1217 (March 2017). DOI: 10.1214/15-AOP1084


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.


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Amir Dembo. Andrea Montanari. Subhabrata Sen. "Extremal cuts of sparse random graphs." Ann. Probab. 45 (2) 1190 - 1217, March 2017.


Received: 1 March 2015; Revised: 1 August 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 1372.05196
MathSciNet: MR3630296
Digital Object Identifier: 10.1214/15-AOP1084

Primary: 05C80 , 68R10 , 82B44

Keywords: bisection , Erdős–Rényi graph , Ising model , Max-cut , Parisi formula , Regular graph , Spin glass

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 2 • March 2017
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