  An exact analytical description of extreme intensity statistics in complex
random states is derived. These states have the statistical properties of the
Gaussian and Circular Unitary Ensemble eigenstates of random matrix theory.
Although the components are correlated by the normalization constraint, it is
still possible to derive compact formulae for all values of the dimensionality
N. The maximum intensity result slowly approaches the Gumbel distribution even
though the variables are bounded, whereas the minimum intensity result rapidly
approaches the Weibull distribution. Since random matrix theory is conjectured
to be applicable to chaotic quantum systems, we calculate the extreme
eigenfunction statistics for the standard map with parameters at which its
classical map is fully chaotic. The statistical behaviors are consistent with
the finite-N formulae.
