  We establish the existence and uniqueness of a well-concentrated giant
component in the supercritical excursion sets of three important ensembles of
spherical Gaussian random fields: Kostlan's ensemble, band-limited ensembles,
and the random spherical harmonics. Our main results prescribe quantitative
bounds for the volume fluctuations of the giant that are essentially optimal
for non-monochromatic ensembles, and suboptimal but still strong for
monochromatic ensembles.
  Our results support the emerging picture that giant components in Gaussian
random field excursion sets have similar large-scale statistical properties to
giant components in supercritical Bernoulli percolation. The proofs employ
novel decoupling inequalities for spherical ensembles which are of independent
interest.
