Application of semi-invariants to proof of the central limit theorem on a lattice

TL;DR

This paper extends the central limit theorem to the Ising model by using semi-invariants to prove that a transformed random field converges to a Gaussian distribution at large scales.

Contribution

It introduces a novel application of semi-invariants to prove a generalized CLT for the Ising model under renormalization.

Findings

Random field converges to Gaussian distribution at large scales

Semi-invariants effectively analyze the thermodynamic limit

Generalization of CLT to lattice-based models

Abstract

Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group transformation. Here we consider a thermodynamic limit of Ising model with weak interaction and we use semi-invariants to prove that a random field transformed by renormalization group converges in distribution to an independent field with Gaussian distribution as the distance scale infinitely increases; it is a generalization of the central limit theorem to the Ising model.