Local Central Limit Theorem for unbounded long-range potentials

TL;DR

This paper establishes the equivalence of integral and local central limit theorems for long-range two-body potentials on lattice systems, using cluster expansion and decimation techniques, applicable to unbounded and bounded spins at various temperatures.

Contribution

It extends the CLT equivalence to unbounded long-range potentials with a novel approach combining cluster expansion and decimation methods.

Findings

Proves CLT equivalence for long-range potentials on lattice systems.

Demonstrates results hold at high temperature for unbounded spins and at all temperatures for bounded spins.

Employs cluster expansion and decimation techniques for bounds and control.

Abstract

We prove the equivalence between the integral central limit theorem and the local central limit theorem for two-body potentials with long-range interactions on the lattice $\mathbb{Z}^d$ for $d\ge 1$. The spin space can be an arbitrary, possibly unbounded subset of the real axis with a suitable a-priori measure. For general unbounded spins, our method works at high-enough temperature, but for bounded spins our results hold for every temperature. Our proof relies on the control of the integrated characteristic function, which is achieved by dividing the integration into three different regions, following a standard approach proposed forty years ago by Campanino, Del Grosso and Tirozzi. The bounds required in the different regions are obtained through cluster-expansion techniques. For bounded spins, the arbitrariness of the temperature is achieved through a decimation ("dilution") technique, also introduced in the later reference.