Universal distribution of random matrix eigenvalues near the "birth of a cut" transition

Abstract

We study the eigenvalue distribution of a random matrix, at a transition where a new connected component of the eigenvalue density support appears away from other connected components. Unlike previously studied critical points, which correspond to rational singularities ρ(x)\sim x^{p/q} classified by conformal minimal models and integrable hierarchies, this transition shows logarithmic and non-analytical behaviours. There is no critical exponent, instead, the power of N changes in a saw teeth behaviour.

arXiv commentLatex, 34 pages, 5 figures

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• Loop Equations and the Topological Phase of Multi-Cut Matrix Models
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• Double scaling limit in the random matrix model: the Riemann-Hilbert approach
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• Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
• All orders asymptotic expansion of large partitions
• A Matrix model for plane partitions
• The birth of a cut in unitary random matrix ensembles
• Phase transitions in multi-cut matrix models and matched solutions of Whitham hierarchies
• An efficient method for computing genus expansions and counting numbers in the Hermitian matrix model
• A new type of critical behaviour in random matrix models
• Universality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schroedinger equation at the first breaking curve
• Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory
• First Colonization of a Spectral Outpost in Random Matrix Theory
• The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the "birth of a cut" transition
• Large N expansions and Painlevé hierarchies in the Hermitian matrix model
• Mesoscopic colonization of a spectral band
• Phase transitions and equilibrium measures in random matrix models
• First colonization of a hard-edge in random matrix theory
• Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit