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.