On the maximum of the potential of a general two-dimensional Coulomb gas

We determine the leading order of the maximum of the logarithmic potential of a two-dimensional Coulomb gas for general β and general confinement potential, extending the result of [LLZ24, Theorem 1]. In particular, the result is new for random normal matrices. The argument relies 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.