Random Matrices in 2D, Laplacian Growth and Operator Theory

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.

88 pages, 8 figures

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Cited by in corpus (20)

• Harmonic moment dynamics in Laplacian growth
• On the exponential transform of multi-sheeted algebraic domains
• The normal matrix model with a monomial potential, a vector equilibrium problem, and multiple orthogonal polynomials on a star
• Selection of the Taylor-Saffman Bubble does not Require Surface Tension
• Laplacian growth in a channel and Hurwitz numbers
• Free Probability for Pairs of Faces II: 2-Variables Bi-free $R$-Transform and Systems with Rank $\le 1$ Commutation
• Stochastic Laplacian growth
• Stream of asymmetric bubbles in a Hele-Shaw channel
• Multiple steadily translating bubbles in a Hele-Shaw channel
• Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics
• Topological transitions in evaporating thin films
• On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth
• Orthogonal polynomials in the normal matrix model with a cubic potential
• Integrability-preserving regularizations of Laplacian Growth
• Lemniscates do not survive Laplacian growth
• Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics
• Variational Methods and Planar Elliptic Growth
• Hamiltonian motions of plane curves and formation of singularities and bubbles
• Multiple bubble dynamics and velocity selection in Laplacian growth without surface tension
• Solitons and Normal Random Matrices