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

作者：Endo, Eric O.^*; Margarint, Vlad

来源：Journal of Statistical Physics, 2022, 189(3): 34.

DOI：10.1007/s10955-022-02994-4

• Local Central Limit Theorem
• Central Limit Theorem
• High Temperatures
• Long-Range Potentials

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.