  Majorana zero modes (MZM-s) occurring at the edges of a 1D, p-wave, spinless
superconductor, in absence of fluctuations of the phase of the superconducting
order parameter, are quintessential examples of topologically-protected
zero-energy modes occurring at the edges of 1D symmetry-protected topological
phases. In this work, we numerically investigate the fate of the topological
phase in the presence of phase-fluctuations using the density matrix
renormalization group (DMRG) technique. To that end, we consider a
one-dimensional array of MZM-s on mesoscopic superconducting islands at zero
temperature. Cooper-pair and MZM-assisted single-electron tunneling, together
with finite charging energy of the mesoscopic islands, give rise to a rich
phase-diagram of this model. We show that the system can be in either a
Mott-insulating phase, a Luttinger liquid (LL) phase of Cooper-pairs or a
second gapless phase. In contrast to the LL of Cooper-pairs, this second phase
is characterized by nonlocal string correlation functions which decay
algebraically due to gapless charge-$e$ excitations. The three phases are
separated from each other by phase-transitions of either Kosterlitz-Thouless or
Ising type. Using a Jordan-Wigner transformation, we map the system to a
generalized Bose-Hubbard model with two types of hopping and use DMRG to
analyze the different phases and the phase-transitions.
