New applications of non-hermitian random matrices

TL;DR

This paper explores the connections between non-Hermitian random matrices and physical phenomena like pattern formation and the quantum Hall effect, revealing new insights into interface dynamics and electronic droplet growth.

Contribution

It uncovers novel links between normal random matrix models and physical systems such as fluid interfaces and quantum Hall droplets, expanding the applications of random matrix theory.

Findings

Large N limit models interface dynamics in fluid systems.

Equivalence between fluid interface growth and electronic droplet expansion.

Matrix models unify different physical phenomena.

Abstract

We discuss recently discovered links of the statistical models of normal random matrices to some important physical problems of pattern formation and to the quantum Hall effect. Specifically, the large $N$ limit of the normal matrix model with a general statistical weight describes dynamics of the interface between two incompressible fluids with different viscousities in a thin plane cell (the Saffman-Taylor problem). The latter appears to be mathematically equivalent to the growth of semiclassical 2D electronic droplets in a strong uniform magnetic field with localized magnetic impurities (fluxes), as the number of electrons increases. The equivalence is most easily seen by relating the both problems to the matrix model.