Generic critical points of normal matrix ensembles

Abstract

The evolution of the degenerate complex curve associated with the ensemble at a generic critical point is related to the finite time singularities of Laplacian Growth. It is shown that the scaling behavior at a critical point of singular geometry $x^3\sim y^2$ is described by the first Painlevé transcendent. The regularization of the curve resulting from discretization is discussed.

Based on a talk given at the conference on Random Matrices, Random Processes and Integrable Systems, CRM Montreal, June 2005

References in corpus (11)

• Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions
• Normal random matrix ensemble as a growth problem
• Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem
• Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime
• Complex Curve of the Two Matrix Model and its Tau-function
• Large scale correlations in normal and general non-Hermitian matrix ensembles
• Partition functions for Matrix Models and Isomonodromic Tau functions
• The solution of a chiral random matrix model with complex eigenvalues
• Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
• The Riemann-Hilbert Problem for the Bi-Orthogonal Polynomials

Cited by in corpus (4)

• Reconstructing 1/2 BPS Space-Time Metrics from Matrix Models and Spin Chains
• Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent
• Integrability-preserving regularizations of Laplacian Growth
• Discrete equations from Bäcklund transformations of the fifth Painlevé equation