  Models of particles driven by a one-dimensional fluctuating surface are known
to exhibit fluctuation dominated phase ordering (FDPO), in which both the order
and fluctuations appear on macroscopic scales. Highly dynamic and
macroscopically broad interfacial regions, each composed of many domain walls,
appear between macroscopically ordered regions and consequently the scaled
correlation function violates the Porod law. We focus on two essential
quantities which together quantify the unique characteristics of FDPO, namely
the total number of domain walls and the length of the largest ordered domain.
We present results in the context of coarse-grained depth (CD) models, both in
steady state and while coarsening. Analytic arguments supported by numerical
simulations show that even though domain wall number fluctuations are very
strong, the associated variance remains constant in time during coarsening.
Further, the length of the largest cluster grows as a power law with
multiplicative logarithms which involve both the time and system size. In
addition, we identify corrections to the leading power law scaling in several
quantities in the coarsening regime. We also study a generalisation of the CD
model in which the domain wall density is controlled by a fugacity and show
that it maps on to the truncated inverse distance squared Ising (TIDSI) model.
The generalised model shows a mixed order phase transition, with the regular CD
model (which exhibits FDPO) corresponding to the critical point.
