  We study some properties of a class of random connected planar fractal sets
induced by a Poissonian scale-invariant and translation-invariant point
process. Using the second-moment method, we show that their Hausdorff
dimensions are deterministic and equal to their expectation dimension. We also
estimate their low-intensity limiting behavior. This applies in particular to
the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops
for which the expectation dimension has been computed by Schramm, Sheffield and
Wilson.
