  We analyze the asymptotic fluctuations of linear eigenvalue statistics of
random centrosymmetric matrices with i.i.d. entries. We prove that for a
complex analytic test function, the centered and normalized linear eigenvalue
statistics of random centrosymmetric matrices converge to a normal
distribution. We find the exact expression of the variance of the limiting
normal distribution via combinatorial arguments. Moreover, we also argue that
the limiting spectral distribution of properly scaled centrosymmetric matrices
follows the circular law.
