Covariance Structure of Coulomb Multiparticle System

We consider a system of particles on a finite interval with Coulomb 3-dimensional interactions between close neighbours, i.e. only a few other neighbours apart. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31-36] to study the flow of charged particles. 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 we include as well interactions beyond the nearest-neighbours ones. Surprisingly but this leads to qualitatively new features even when the external force is zero. The order of the covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case, and moreover the covariances exhibit periodicity in sign: the interspacings are positively correlated if the number of interspacings between them is odd, otherwise, they are negatively correlated. In the course of the proof we derive Gaussian approximation for the limit distribution for dependent variables described by a Gibbs distribution.Comment: 43 page

Exponential decay of correlations in the one-dimensional Coulomb gas ensembles

We consider the Gibbs measure on the configurations of N particles on R+ with one fixed particle at one end at 0. The potential includes pair-wise Coulomb interactions between any particle and its 2K neighbors. Only when K = 1, the model is within the rank-one operators, and it was treated previously. Here, for the case K ≥ 2, exponentially fast convergence of density distribution for the spacings between particles is proved when N → ∞. In addition, we establish the exponential decay of correlations between the spacings when the number of particles between them is increasing. We treat in detail the case K = 2; when K > 2, the proof works in a similar manner

Phase transitions in the one-dimensional coulomb gas ensembles

We consider the system of particles on a finite interval with pairwise nearest neighbours interaction and external force. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31–36] to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states). We derive the asymptotics for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity. We identify 5 different phases for any positive temperature. The proofs rely on a conditional central limit theorem for nonidentical random variables, which has an interest on its own