  Bose-Einstein condensates made of ultracold trapped bosonic atoms have become
a central venue in which interacting many-body quantum systems are studied. The
ground state of a trapped Bose-Einstein condensate has been proven to be 100%
condensed in the limit of infinite particle number and constant interaction
parameter [Lieb and Seiringer, Phys. Rev. Lett. {\bf 88}, 170409 (2002)]. The
meaning of this result is that properties of the condensate, noticeably its
energy and density, converge to those obtained by minimizing the
Gross-Pitaevskii energy functional. This naturally raises the question whether
correlations are of any importance in this limit. Here, we demonstrate both
analytically and numerically that even in the infinite particle limit many-body
correlations can lead to a substantial modification of the \textit{variance} of
any operator compared to that expected from the Gross-Pitaevskii result. The
strong deviation of the variance stems from its explicit dependence on terms of
the reduced two-body density matrix which otherwise do not contribute to the
energy and density in this limit. This makes the variance a sensitive probe of
many-body correlations even when the energy and density of the system have
already converged to the Gross-Pitaevskii result. We use the center-of-mass
position operator to exemplify this persistence of correlations. Implications
of this many-body effect are discussed.
