  Using the proposed by us thinning approach to describe extreme matrices, we
find an explicit exponentiation formula linking classical extreme laws of
Fr\'echet, Gumbel and Weibull given by Fisher-Tippet-Gnedenko classification
and free extreme laws of free Fr\'echet, free Gumbel and free Weibull by Ben
Arous and Voiculescu [1]. We also develop an extreme random matrix formalism,
in which refined questions about extreme matrices can be answered. In
particular, we demonstrate explicit calculations for several more or less known
random matrix ensembles, providing examples of all three free extreme laws.
Finally, we present an exact mapping, showing the equivalence of free extreme
laws to the Peak-Over-Threshold method in classical probability.
