  We establish probabilistic small data global well-posedness of the
energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for
scaling super-critical random initial data. The proof relies on an induction on
frequency procedure and a modified linear-nonlinear decomposition furnished by
a delicate "probabilistic" parametrix construction. This is the first global
existence result for a geometric wave equation for random initial data at
scaling super-critical regularity.
