  We continue our investigation of the fractal uncertainty principle (FUP) for
random fractal sets. In the prequel (arXiv:2107.08276), we considered the
Cantor sets in the discrete setting with alphabets randomly chosen from a base
of digits so the dimension d is in (0,2/3). We proved that, with overwhelming
probability, the FUP with an exponent >=1/2-3d/4- holds for these discrete
Cantor sets with random alphabets.
  In this sequel, we construct random Cantor sets with dimension d in (0,2/3)
in R via a different random procedure from the one in the prequel. We prove
that, with overwhelming probability, the FUP with an exponent >=1/2-3d/4-
holds. The proof follows from establishing a Fourier decay estimate of the
corresponding random Cantor measures, which is in turn based on a concentration
of measure phenomenon in an appropriate probability space for the random Cantor
sets.
