  The fractal dimensions of the hull, the external perimeter and of the red
bonds are measured through Monte Carlo simulations for dilute minimal models,
and compared with predictions from conformal field theory and SLE methods. The
dilute models used are those first introduced by Nienhuis. Their loop fugacity
is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their
description through conformal loop ensembles. It is also linked to conformal
field theories through their central charges c = 13 - 6(barkappa +
barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p'
where p and p' are two coprime integers. The geometric exponents of the hull
and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4),
(4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo
upgrades are proposed for these models as well as several techniques to improve
their speeds. The measured fractal dimensions are obtained by extrapolation on
the lattice size H,V -> infinity. The extrapolating curves have large slopes;
despite these, the measured dimensions coincide with theoretical predictions up
to three or four digits. In some cases, the theoretical values lie slightly
outside the confidence intervals; explanations of these small discrepancies are
proposed.
