  We investigate Josephson currents in mesoscopic rings with a weak link which
are in or near a topological superconducting phase. As a paradigmatic example,
we consider the Kitaev model of a spinless p-wave superconductor in one
dimension, emphasizing how this model emerges from more realistic settings
based on semiconductor nanowires. We show that the flux periodicity of the
Josephson current provides signatures of the topological phase transition and
the emergence of Majorana fermions situated on both sides of the weak link even
when fermion parity is not a good quantum number. In large rings, the Majorana
fermions hybridize only across the weak link. In this case, the Josephson
current is h/e periodic in the flux threading the loop when fermion parity is a
good quantum number but reverts to the more conventional h/2e periodicity in
the presence of fermion-parity changing relaxation processes. In mesoscopic
rings, the Majorana fermions also hybridize through their overlap in the
interior of the superconducting ring. We find that in the topological
superconducting phase, this gives rise to an h/e-periodic contribution even
when fermion parity is not conserved and that this contribution exhibits a peak
near the topological phase transition. This signature of the topological phase
transition is robust to the effects of disorder. As a byproduct, we find that
close to the topological phase transition, disorder drives the system deeper
into the topological phase. This is in stark contrast to the known behavior far
from the phase transition, where disorder tends to suppress the topological
phase.
