On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas

We determine the leading order of the maximum of the random potential associated to a two-dimensional Coulomb gas for general β and general confinement potential, extending the recent result of Lambert-Lebl\'e-Zeitouni. In the case β=2, this corresponds to the (centered) log-characteristic polynomial of either the Ginibre random matrix ensemble for V(x)=|x|²/2 or a more general normal matrix ensemble. The result on the leading order asymptotics for the maximum of the log-characteristic polynomial is new for random normal matrices. We rely on connections with the classical obstacle problem and the theory of Gaussian Multiplicative Chaos. We make use of a new concentration result for fluctuations of C1,1 linear statistics which may be of independent interest.