Quantum Hele-Shaw flow

TL;DR

This paper explores the quantum Hele-Shaw flow, a process in the complex plane related to electronic droplets and random matrices, extending previous results and showing convergence to classical flow under certain conditions.

Contribution

It extends existing results to general potentials and proves convergence of quantum to classical Hele-Shaw flow for smooth potentials.

Findings

Extended results to general external field potentials.

Proved convergence to classical Hele-Shaw flow for smooth potentials.

Connected quantum flow to obstacle problem in classical flow modeling.

Abstract

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.