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We study the Gross-Pitaevskii (GP) energy functional for a fast rotatingBose-Einstein condensate on the unit disc in two dimensions. Writing thecoupling parameter as $ 1 / \eps^2 $ we consider the asymptotic regime $ \eps\to 0 $ with the angular velocity $\Omega$ proportional to $(\eps^2|\log\eps|)^{-1} $. We prove that if $ \Omega = \Omega_0(\eps^2|\log\eps|)^{-1} $ and $ \Omega_0 > 2(3\pi)^{-1} $ then a minimizer ofthe GP energy functional has no zeros in an annulus at the boundary of the discthat contains the bulk of the mass. The vorticity resides in a complementary`hole' around the center where the density is vanishingly small. Moreover, weprove a lower bound to the ground state energy that matches, up to smallerrors, the upper bound obtained from an optimal giant vortex trial function,and also that the winding number of a GP minimizer around the disc is in accordwith the phase of this trial function.

The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate