  We introduce a simple model of touching random surfaces, by adding a chemical
potential rho for ``minimal necks'', and study this model numerically coupled
to a Gaussian model in d-dimensions (for central charge c = d = 0, 1 and 2).
For c <= 1, this model has a phase transition to branched polymers, for
sufficiently large rho. For c = 2, however, the extensive simulations indicate
that this transition is replaced by a cross-over behavior on finite lattices
--- the model is always in the branched polymer phase. This supports recent
speculations that, in 2d-gravity, the behavior observe in simulations for $c
\leq 1$, is dominated by finite size effects, which are exponentially enhanced
as c -> 1+.
