Block Toeplitz matrices and the two-dimensional Coulomb gas near a wall

For the two-dimensional Coulomb gas on a lattice, at the special value of the dimensionless coupling constant=2, the grand partition function and correlations can be written in terms of the eigenvalues and eigenvectors of a block Toeplitz matrix. By using the semiperiodic Coulomb potential and taking the continuum limit in the periodic direction so as to have a set of parallel lines as the domain, it is shown that these eigenvalues and eigenvectors can be computed exactly. This allows the pressure and the correlations near a charged wall to be rigorously evaluated. The two-particle correlations obey a sum rule which implies that the state in the vicinity of the wall is a conductor.