  Dimensionality strongly affects thermal fluctuations and critical dynamics of
equilibrium systems. These influences persist in amorphous systems going
through the nonequilibrium glass transition. Here, we experimentally study the
glass transition of quasi-2D suspensions of spherical and ellipsoidal particles
under different degrees of circular confinement. We show that the strength of
the long-wavelength fluctuations increases logarithmically with system sizes
and displays the signature of the Mermin-Wagner fluctuations. Moreover, using
confinement as a tool, we also measure static structural correlations and
extract a growing static correlation length in 2D supercooled liquids. Finally,
we explore the influence of the Mermin-Wagner fluctuations on the translational
and orientational relaxations of 2D ellipsoidal suspensions, which leads to a
new interpretation of the two-step glass transition and the orientational glass
phase of anisotropic particles. Our study reveals the importance of
long-wavelength fluctuations in 2D supercooled liquids and provides new
insights into the role of dimensionality in the glass transition.
