  We present measurements of the fractal dimensions associated to the
geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure
similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters
in these models but we discovered that these clusters do not percolate at the
critical point of the model under consideration. These results clearly mark a
difference in the behaviour of these non local objects compared to the Ising
model or the 3-state Potts model which corresponds to the simplest cases of Z_N
spin models with N=2 and N=3 respectively. We compare these fractal dimensions
with the ones obtained for SLE interfaces.
