Antiferromagnetic Covariance Structure of Coulomb Chain

We consider a system of particles lined up on a finite interval in a 3-dimensional space with Coulomb interactions between the nearest and next to the nearest neighbours. This model was introduced by Malyshev (Probl Inf Transm 51(1):31–36, 2015) to study the flow of charged particles. The distribution of spacings between the consecutive particles is of interest. Notably, even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here, assuming zero external force, we show that interactions beyond the nearest ones lead to qualitatively new features of the system. In particular, the order of decay (in terms of the total number of particles) of covariances between the spacings is changed when compared with the former nearest-neighbours case. Furthermore, we discover that the covariances between spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of the number of spacings between them, while their amplitude decays. In the course of the proof of these results, a conditional Central Limit Theorem for dependent random variables is established.