Universal shocks in random matrix theory

Abstract

We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the case of the Gaussian Unitary Ensemble, on which we focus in this letter, we show that the orthogonal polynomials, and their Cauchy transforms, evolve according to a viscid Burgers equation with an effective "spectral viscosity" ${\nu }_s=1/2N$, where $N$ is the size of the matrices. We relate the edge of the spectrum of eigenvalues to the shock that naturally appears in the Burgers equation for appropriate initial conditions, thereby obtaining a new perspective on universality.

4 pages, no figures

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• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Full Dysonian dynamics of the complex Ginibre ensemble
• Non-analyticity in scale in the planar limit of QCD
• Universal shocks in the Wishart random matrix ensemble - 1
• Stability of large complex systems with heterogeneous relaxation dynamics
• Eigenvalue density of Wilson loops in 2D SU(N) YM
• Hydrodynamic Limit of Multiple SLE
• Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD
• Handbook Article on Applications of Random Matrix Theory to QCD
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Current fluctuations in the Dyson Gas
• Eikonal formulation of large dynamical random matrix models
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• Quasi-Normal Modes from Non-Commutative Matrix Dynamics
• Hydrodynamics of the Chiral Dirac Spectrum
• Characterizations of the hydrodynamic limit of the Dyson model