Universal shocks in the Wishart random-matrix ensemble - a sequel

Abstract

We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.

7 pages, 2 figures

References in corpus (8)

• The Pearcey Process
• Exact distribution of the maximal height of p vicious walkers
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie
• Non-intersecting squared Bessel paths: critical time and double scaling limit
• The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
• Universal shocks in the Wishart random matrix ensemble - 1
• Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD

Cited by in corpus (7)

• Dysonian dynamics of the Ginibre ensemble
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Full Dysonian dynamics of the complex Ginibre ensemble
• Chiral Random Matrix Model at Finite Chemical Potential: Characteristic Determinant and Edge Universality
• Three-Parametric Marcenko-Pastur Density
• Functional Equations Solving Initial-Value Problems of Complex Burgers-Type Equations for One-Dimensional Log-Gases