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arXiv · 1306.4014

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

Jean-Paul Blaizot·Maciej A. Nowak·Piotr Warchoł
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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.

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Jean-Paul Blaizot, Maciej A. Nowak, Piotr Warchoł. 2013-09-04. Universal shocks in the Wishart random-matrix ensemble - a sequel. https://doi.org/10.1103/physreve.89.042130

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