Given a Riemann surface X=(Σ,J), we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form (0,0,⋯,tωn). Moreover, we find an associated equivariant map from the universal cover (Σ~,J~) to the symmetric space SLn(C)/SU(n) and show that limits of these maps tend to a sub-building in the asymptotic cone. That sub-building is explicitly constructed from the local data of ωn.