The "Screening Phase Transitions" in the Two Dimensional Coulomb Gas

The Mayer series of a Coulomb gas with fixed ultraviolet cutoff is studied in two dimensions. In particular, we show the existence of infinitely many thresholds $k T_n =(e^2/8\pi k)(1-1/2n)^{–1}$ k=Boltzmann's constant,e=electric charge, $n=1, 2,...$, which are conjectured to reflect a sequence of transitions from pure multipole phase (the Kosterlitz-Thouless region) to a plasma phase (the Debye screening region) via an infinite number of "intermediate phases". Mathematically we prove that the Mayer series' coefficients of order up to $2n$ are finite if the temperature $T$ is $< T_n$. For $T