  A bosonic analogue of the fractional quantum Hall eff ect occurs in rapidly
rotating trapped Bose gases: There is a transition from uncorrelated Hartree
states to strongly correlated states such as the Laughlin wave function. This
physics may be described by eff ective Hamiltonians with delta interactions
acting on a bosonic N-body Bargmann space of analytic functions. In a previous
paper [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)] we
studied the case of a quadratic plus quartic trapping potential and derived
conditions on the parameters of the model for its ground state to be
asymptotically strongly correlated. This relied essentially on energy upper
bounds using quantum Hall trial states, incorporating the correlations of the
Bose-Laughlin state in addition to a multiply quantized vortex pinned at the
origin. In this paper we investigate in more details the density of these trial
states, thereby substantiating further the physical picture described in [N.
Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)], improving
our energy estimates and allowing to consider more general trapping potentials.
Our analysis is based on the interpretation of the densities of quantum Hall
trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy).
New estimates on the mean- field limit of such systems are presented.
