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Rigorous Results for Hierarchical Models of Structural Glasses

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Abstract

We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model, and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the \(p\)-spin model of a structural glass, and we prove the existence of the thermodynamic limit and self-averaging of the free energy.

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Acknowledgments

We would like to thank A. Barra and F. Guerra for useful discussions. Research supported by NSF Grants PHY–0957573, by the Human Frontiers Science Program, by the Swartz Foundation, and by the W. M. Keck Foundation.

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  1. Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, 08544, USA

    Michele Castellana

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  1. Michele Castellana
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Correspondence to Michele Castellana.

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Castellana, M. Rigorous Results for Hierarchical Models of Structural Glasses. J Stat Phys 157, 219–233 (2014). https://doi.org/10.1007/s10955-014-1085-9

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