  As part of our ongoing work on the enumeration of symmetry classes of lozenge
tilings of hexagons with certain four-lobed structures removed from their
center, we consider the case of the tilings which are both vertically and
horizontally symmetric. In order to handle this, we need an extension of Kuo's
graphical condensation method, which works in the presence of free boundary.
Our results allow us to compute exactly the correlation in a sea of dimers of a
macroscopic dent in a 90 degree wedge with mixed boundary conditions. We use
previous results to compute the correlation of the corresponding symmetrized
system with no boundary, and show that its fourth root has the same
log-asymptotics as the correlation of the dent in the 90 degree wedge. This is
the first result of this kind involving a macroscopic defect. It suggests that
the connections between dimer systems with gaps and 2D electrostatics may be
deeper that previously thought.
