Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions.
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, i.e., in the vicinity of a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.
IPTh, CNRS, URA No. 2306, CEA Saclay, 91191 Gif-sur-Yvette, France.
Jean-Paul
JP
Blaizot
M. Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University, PL-30-059 Cracow, Poland.
Maciej A
MA
Nowak
M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30-059 Cracow, Poland.
Piotr
P
Warchoł