Charge Fluctuations for a Coulomb Fluid in a Disk on a Pseudosphere

The classical (i.e. non-quantum) equilibrium statistical mechanics of a Coulomb fluid living on a pseudosphere (an infinite surface of constant negative curvature) is considered. The Coulomb fluid occupies a large disk communicating with a reservoir (grand-canonical ensemble). The total charge $Q$ on the disk fluctuates. In a macroscopic description, the charge correlations near the boundary circle can be described as correlations of a surface charge density $\sigma$. In a macroscopic approach, the variance of $Q$ and the correlation function of $\sigma$ are computed; they are universal. These macroscopic results are shown to be valid for two solvable microscopic models, in the limit when the microscopic thickness of the surface charge density goes to zero.