Quantum Hele-Shaw flow

In this note, we discuss the quantum Hele-Shaw flow, a random measure process in the complex plane introduced by the physicists P.Wiegmann, A. Zabrodin, et al. This process arises in the theory of electronic droplets confined to a plane under a strong magnetic field, as well as in the theory of random normal matrices. We extend a result of Elbau and Felder to general external field potentials, and also show that if the potential is $C^2$-smooth, then the quantum Hele-Shaw flow converges, under appropriate scaling, to the classical (weighted) Hele-Shaw flow, which can be modeled in terms of an obstacle problem.