Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition

We study the distribution of the mean radial displacement of charges of a 2D one-component plasma in the thermodynamic limit $N\to \mathrm{\infty }$ at finite temperature $\beta >0$. We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at the ground state of the plasma. These results have been compared with the integrable case (finite $N$) of plasma parameter $\beta q^2=2$. In this case the problem can be mapped to the stationary properties of 2D Dyson Brownian particles and to a non-Hermitian matrix model.