Rotated semicircle laws for permanental roots of Gaussian random matrices

Abstract

For matrices drawn from the standard Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE), we prove that the normalized zero counting measure of the permanental characteristic polynomial $Per(zI_N-H_N)$ converges almost surely to the standard Wigner semicircle law on $[-2,2]$, rotated by $\pi/2$ onto the imaginary axis. This proves the conjecture proposed by Fyodorov in \cite{Fyodorov2006}.