Critical exponents of the two dimensional Coulomb gas at the Berezinskii-Kosterlitz-Thouless transition

The two dimensional Coulomb gas is the prototypical model of statistical mechanics displaying a special kind of phase transition, named after Berezinskii, Kosterlitz and Thouless. Physicists and mathematicians proposed several predictions about this system. Two of them, valid along the phase transition curve and for small activity, are: a) the long-distance decay of the "fractional charge" correlation is power law, with a multiplicative logarithmic correction; b) in such a decay, the exponent of the power law, as well as the exponent of the logarithmic correction, have a certain precise dependence upon the charge value. In this paper we provide a proof of these two long standing conjectures.