  We consider the gelation of particles which are permanently connected by
random crosslinks, drawn from an ensemble of finite-dimensional continuum
percolation. To average over the randomness, we apply the replica trick, and
interpret the replicated and crosslink-averaged model as an effective molecular
fluid. A Mayer-cluster expansion for moments of the local static density
fluctuations is set up. The simplest non-trivial contribution to this series
leads back to mean-field theory. The central quantity of mean-field theory is
the distribution of localization lengths, which we compute for all
connectivities. The highly crosslinked gel is characterized by a one-to-one
correspondence of connectivity and localization length. Taking into account
higher contributions in the Mayer-cluster expansion, systematic corrections to
mean-field can be included. The sol-gel transition shifts to a higher number of
crosslinks per particle, as more compact structures are favored. The critical
behavior of the model remains unchanged as long as finite truncations of the
cluster expansion are considered. To complete the picture, we also discuss
various geometrical properties of the crosslink network, e.g. connectivity
correlations, and relate the studied crosslink ensemble to a wider class of
ensembles, including the Deam-Edwards distribution.
