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Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems

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Abstract

In two papers Franz et al. proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution (Franz and Leone in J Stat Phys 111(3–4):535–564, 2003) and a general distribution (Franz et al. in J Phys A 36(43), 10967, 2003). Panchenko and Talagrand (Probab Theo Relat Fields 130(3):319–336, 2004) simplified the proof and generalized the result of Franz and Leone (J Stat Phys 111(3–4):535–564, 2003) for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks (Salez in Combin Probab Comput 25(03):436–447, 2016) and allows to bypass the arguments in Franz et al. (J Phys A 36(43):10967, 2003) based on the study of the Sherrington–Kirkpatrick (SK) model.

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Authors and Affiliations

  1. Inria - École Normale Supérieure, Paris, France

    Marc Lelarge

  2. École Normale Supérieure, Paris, France

    Mendes Oulamara

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  1. Marc Lelarge
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  2. Mendes Oulamara
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Correspondence to Mendes Oulamara.

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Lelarge, M., Oulamara, M. Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems. J Stat Phys 173, 917–940 (2018). https://doi.org/10.1007/s10955-018-1964-6

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  • DOI: https://doi.org/10.1007/s10955-018-1964-6

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