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

TL;DR

This paper proves that at high temperatures, the Local Central Limit Theorem holds for certain long-range pair potentials in Gibbs measures, extending previous results to more general conditions and state spaces.

Contribution

It demonstrates that the Local Central Limit Theorem follows from the Integral Central Limit Theorem for long-range potentials at high temperatures under broader conditions.

Findings

Local CLT holds for long-range potentials at high temperatures

Extension to general state spaces with finite measures

Broader conditions beyond previous work

Abstract

Dobrushin and Tirozzi [14] 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, Capocaccia, and Tirozzi [7] 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 [7], 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 [7] when the state space is general, provided that it is equipped with a finite measure.