The Kosterlitz-Thouless transitions by the mayer expansion finiteness of the perturbations

The Kosterlitz-Thouless transitions in 2D XY and Coulomb gas models are discussed by the Mayer expansion. After transforming the 2D XY model into a (generalzed) 2D Coulomb gas model by the duality transformation, it is shown that the free energy (pressure) and the two point correlation function 〈cos(θ₀?θ_ζ)〉 are expressed as Σa_N(β) and $\text{exp}\text{[(}\text{2}\text{β}\text{) C}{}_0\text{(ζ) + Σb}{}_{\mathrm{N}}\text{(β:ζ)]}$, respectively, for large inverse temperature β > β_c, where {a_N(β)} are the usual virial coefficients and {a_N, b_N} are the contributions from N-electron system. Moreover C₀(ζ) ? ? (2π)^?1 log(|ζ| + 1), |a_N(β)| ≤ C₁(N) exp[?βK₃N] and |b_N(β:N) ≤ C₂(N) exp[?βK₄N] |C₀(ζ)| (K_i > 0). By comparing this system with the dipole gas system in which these series converge absolutely, it is conjectured that these series converge absolutely for large β.