Universal shocks in the Wishart random matrix ensemble - 1

Abstract

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution through the method of characteristics presents singularities that we relate to the precursors of shock formation in fluid dynamical equations. The 1/N corrections may be viewed as viscous corrections, with the role of the viscosity being played by the inverse of the doubled dimension of the matrix. These corrections are studied through a scaling analysis in the vicinity of the shocks, and one recovers in a simple way the universal Bessel oscillations (so-called hard edge singularities) familiar in random matrix theory.

9 pages

References in corpus (5)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Large N_c confinement and turbulence
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Burgers' equation in 2D SU(N) YM
• Complex Burgers' equation in 2D SU(N) YM

Cited by in corpus (12)

• Dysonian dynamics of the Ginibre ensemble
• Raney distributions and random matrix theory
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Full Dysonian dynamics of the complex Ginibre ensemble
• Hydrodynamical spectral evolution for random matrices
• Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD
• Diffusion method in Random Matrix Theory
• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Eikonal formulation of large dynamical random matrix models
• Hydrodynamics of the Chiral Dirac Spectrum
• Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases
• Three-Parametric Marcenko-Pastur Density