We study the stability of nucleated topological phases that can emerge when interacting non-Abelian anyons form a regular
array. The studies are carried out in the context of Kitaev's honeycomb model, where we consider three distinct types of perturbations
in the presence of a lattice of Majorana mode binding vortices—spatial anisotropy of the vortices, dimerization of the vortex
lattice, and local random disorder. While all the nucleated phases are stable with respect to weak perturbations of each kind,
strong perturbations are found to result in very different behavior. Anisotropy of the vortices stabilizes the strong-pairing-like
phases, while dimerization can recover the underlying non-Abelian phase. Local random disorder, on the other hand, can drive
all the nucleated phases into a gapless thermal metal state. We show that all these distinct behaviors can be captured by
an effective staggered tight-binding model for the Majorana modes. By studying the pairwise interactions between the vortices,
i.e., the amplitudes for the Majorana modes to tunnel between vortex cores, the locations of phase transitions and the nature
of the resulting states can be predicted. We also find that, due to oscillations in the Majorana tunneling amplitude, lattices
of Majorana modes may exhibit a Peierls-like instability, where a dimerized configuration is favored over a uniform lattice.
As the nature of the nucleated phases depends only on the Majorana tunneling, our results are expected to apply also to other
system supporting localized Majorana mode arrays, such as Abrikosov lattices in p-wave superconductors, Wigner crystals in
Moore-Read fractional quantum Hall states, or arrays of topological nanowires.