Convergence of the spectral radius of a random matrix through its characteristic polynomial

Consider a square random matrix with independent and identically distributed entries of mean zero unit variance. We show that as the dimension tends to infinity, spectral radius is equivalent root in probability. This result can also be seen convergence support circular law theorem under optimal moment conditions. In proof we establish reciprocal characteristic polynomial analytic function outside disc, related hyperbolic Gaussian function. The short differs from usual approaches for radius. It relies on tightness argument joint central limit phenomenon traces fixed powers.