  We consider operators in ${\cal N}=4$ super Yang-Mills theory dual to closed
string states propagating on a class of LLM geometries. The LLM geometries we
consider are specified by a boundary condition that is a set of black rings on
the LLM plane. When projected to the LLM plane, the closed strings are polygons
with all corners lying on the outer edge of a single ring. The large $N$ limit
of correlators of these operators receives contributions from non-planar
diagrams even for the leading large $N$ dynamics. Our interest in these
fluctuations is because a previous weak coupling analysis argues that the net
effect of summing the huge set of non-planar diagrams, is a simple rescaling of
the 't Hooft coupling. We carry out some nontrivial checks of this proposal.
Using the $su(2|2)^2$ symmetry we determine the two magnon $S$-matrix and
demonstrate that it agrees, up to two loops, with a weak coupling computation
performed in the CFT. We also compute the first finite size corrections to both
the magnon and the dyonic magnon by constructing solutions to the Nambu-Goto
action that carry finite angular momentum. These finite size computations
constitute a strong coupling confirmation of the proposal.
