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