  We prove that the tagged particles of infinitely many Brownian particles in $
\Rtwo $ interacting via a logarithmic (two-dimensional Coulomb) potential with
inverse temperature $ \beta = 2 $ are sub-diffusive. The associated delabeled
diffusion is reversible with respect to the Ginibre random point field, and the
dynamics are thus referred to as the Ginibre interacting Brownian motion. % If
the interacting Brownian particles have interaction potential $ \Psi $ of
Ruelle class and the total system starts in a translation-invariant equilibrium
state, then the tagged particles are always diffusive if the dimension $ \dd $
of the space $ \mathbb{R}^{\dd } $ is greater than or equal to two. That is,
the tagged particles are always non-degenerate under diffusive scaling. Our
result is, therefore, contrary to known results. The Ginibre random point field
has various levels of geometric rigidity. Our results reveal that the geometric
property of infinite particle systems affects the dynamical property of the
associated stochastic dynamics.
