  Mixed order transitions are those which show a discontinuity of the order
parameter as well as a divergent correlation length. We show that the behaviour
of the order parameter correlation function along the transition line of mixed
order transitions can change from normal critical behaviour with power law
decay, to fluctuation-dominated phase ordering as a parameter is varied. The
defining features of fluctuation-dominated order are anomalous fluctuations
which remain large in the thermodynamic limit, and correlation functions which
approach a finite value through a cusp singularity as the separation scaled by
the system size approaches zero. We demonstrate that fluctuation-dominated
order sets in along a portion of the transition line of an Ising model with
truncated long-range interactions which was earlier shown to exhibit mixed
order transitions, and also argue that this connection should hold more
generally.
