  We introduce a family of boundary confinements for Coulomb gas ensembles, and
study them in the two-dimensional determinantal case of random normal matrices.
The family interpolates between the free boundary and hard edge cases, which
have been well studied in various random matrix theories. The confinement can
also be relaxed beyond the free boundary to produce ensembles with more fuzzy
boundaries, i.e., where the particles are more and more likely to be found
outside of the boundary. The resulting ensembles are investigated with respect
to scaling limits and distribution of the maximum modulus. In particular, we
prove existence of a new point field - a limit of scaling limits to the
ultraweak point when the droplet ceases to be well defined.
