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Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures

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Abstract

Dobrushin and Tirozzi (Commun Math Phys 54(2):173–192, 1977) showed that, for a Gibbs measure with the finite-range potential, the Local Central Limit Theorem is implied by the Integral Central Limit Theorem. Campanino et al. (Commun Math Phys 70(2):125–132, 1979) extended this result for a family of Gibbs measures for long-range pair potentials satisfying certain conditions. We are able to show for a family of Gibbs measures for long-range pair potentials not satisfying the conditions given in Campanino et al. (Commun Math Phys 70(2):125–132, 1979) , that at sufficiently high temperatures, if the Integral Central Limit Theorem holds for a given sequence of Gibbs measures, then the Local Central Limit Theorem also holds for the same sequence. We also extend (Campanino et al. in Commun Math Phys 70(2):125–132, 1979) to the case when the state space is general, provided that it is equipped with a finite measure.

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Acknowledgements

The authors would like to kindly thank Roberto Fernández and Tong Xuan Nguyen for very useful discussions. We thank the referees and Aernout van Enter for their suggestions that helped us clarify our manuscript. Also, both authors would like to acknowledge the support of the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.

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No funds, grants, or other support was received beyond acknowledgements of NYU Shanghai. The authors have no relevant financial or non-financial interests to disclose.

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  1. Eric O. Endo and Vlad Margarint have contributed equally to this work.

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  1. NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China

    Eric O. Endo & Vlad Margarint

  2. University of Colorado Boulder, 2300 Colorado Avenue, Boulder, 80309-0395, USA

    Vlad Margarint

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  1. Eric O. Endo
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Endo, E.O., Margarint, V. Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures. J Stat Phys 189, 34 (2022). https://doi.org/10.1007/s10955-022-02994-4

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