  The Max-Min and Min-Max of matrices arise prevalently in science and
engineering. However, in many real-world situations the computation of the
Max-Min and Min-Max is challenging as matrices are large and full information
about their entries is lacking. Here we take a statistical-physics approach and
establish limit-laws -- akin to the Central Limit Theorem -- for the Max-Min
and Min-Max of large random matrices. The limit-laws intertwine random-matrix
theory and extreme-value theory, couple the matrix-dimensions geometrically,
and assert that Gumbel statistics emerge irrespective of the matrix-entries'
distribution. Due to their generality and universality, as well as their
practicality, these novel results are expected to have a host of applications
in the physical sciences and beyond.
