Random Matrices in 2D, Laplacian Growth and Operator Theory

TL;DR

This paper reviews recent advances in two-dimensional and fractal random matrix theory, highlighting its applications in physical problems and extending classical one-dimensional results to higher dimensions.

Contribution

It introduces recent developments in 2D and fractal random matrix theory and discusses their potential applications in physics.

Findings

Extension of RMT to 2D and fractal spectra

Connections between RMT and Laplacian growth phenomena

Potential applications in physical systems

Abstract

Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of theoretical physics, from condensed matter to high energy. The fundamental results obtained so far rely mostly on the theory of random matrices in one dimension (the dimensionality of the spectrum, or equilibrium probability density). In the last few years, this theory has been extended to the case where the spectrum is two-dimensional, or even fractal, with dimensions between 1 and 2. In this article, we review these recent developments and indicate some physical problems where the theory can be applied.