Abstract
We study rigorously a lattice gas version of the Sherrington–Kirckpatrick spin glass model. In discrete optimization literature this problem is known as unconstrained binary quadratic programming and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present a heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances.
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Acknowledgements
The authors are greateful to Pierre Picco for many fruitful discussions and to Fabio Lucio Toninelli for his kind help. AT thanks the Department of Mathematics of the University of Rome “Tor Vergata” for the support received and for the warm hospitality. Our work has been partially supported by PRIN 2012, Problemi matematici in teoria cinetica ed applicazioni. BS thanks the support of the A*MIDEX Project (n. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR). The authors want to thank the anonymous referees for their precious work.
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Scoppola, B., Troiani, A. Gaussian Mean Field Lattice Gas. J Stat Phys 170, 1161–1176 (2018). https://doi.org/10.1007/s10955-018-1984-2
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DOI: https://doi.org/10.1007/s10955-018-1984-2