title: Universal shocks in the Wishart random-matrix ensemble.
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 inviscid 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 the Burgers equation. The finite N effects appear as a viscosity term in the Burgers equation. Using a scaling analysis of the complete equation for the characteristic polynomial, in the vicinity of the shocks, we recover in a simple way the universal Bessel oscillations (so-called hard-edge singularities) familiar in random-matrix theory.
authors.affiliation: Institut de Physique Théorique (IPhT), CNRS/URA 2306, CEA-Saclay, 91191 Gif-sur Yvette, France. Jean-Paul.Blaizot@cea.fr
authors.forename: Jean-Paul
authors.initials: JP
authors.lastname: Blaizot
authors.forename: Maciej A
authors.initials: MA
authors.lastname: Nowak
authors.forename: Piotr
authors.initials: P
authors.lastname: Warchoł
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D015233:Models, Statistical; D009716:Numerical Analysis, Computer-Assisted