  The critical properties displayed by an ideal 2D Bose gas trapped in a
harmonic potential are determined and characterized in an exact numerical
fashion. Beyond thermodynamics, addressed in terms of the global pressure and
volume which are the appropriate variables of a fluid confined in a non-uniform
harmonic potential, the density-density correlation function is also calculated
and the corresponding correlation length is found. Evaluation of all these
quantities as Bose-Einstein condensation (BEC) is approached manifest its
critical continuous phase transition character. The divergence of the
correlation length as the critical temperature is reached, unveils the expected
spatial scale invariance proper of a critical transition. The logarithmic
singularities of this transition are traced back to the non-analytic behavior
of the thermodynamic variables at vanishing chemical potential, which is the
onset of BEC. The critical exponents associated with the ideal BEC transition
in the 2D inhomogeneous fluid reveals its own universality class.
