Quantum Hele-Shaw flow

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.

24 pages

References in corpus (3)

• Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions
• Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime
• Density of eigenvalues of random normal matrices

Cited by in corpus (7)

• Fluctuations of eigenvalues of random normal matrices
• Coulomb gas ensembles and Laplacian growth
• Edge scaling limits for a family of non-Hermitian random matrix ensembles
• Convergence of Bergman measures for high powers of a line bundle
• Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices
• Asymptotic Integral Kernel for Ensembles of Random Normal Matrix with Radial Potentials
• Conformal Deformation from Normal to Hermitian Random Matrix Ensembles