  The coagulation (or aggregation) equation was introduced by Smoluchowski in
1916 to describe the clumping together of colloidal particles through
diffusion, but has been used in many different contexts as diverse as physical
chemistry, chemical engineering, atmospheric physics, planetary science, and
economics. The effectiveness of clumping is described by a kernel $K(x,y)$,
which depends on the sizes of the colliding particles $x,y$. We consider
kernels $K = (xy)^{\gamma}$, but any homogeneous function can be treated using
our methods. For sufficiently effective clumping $1 \ge \gamma > 1/2$, the
coagulation equation produces an infinitely large cluster in finite time (a
process known as the gel transition). Using a combination of analytical methods
and numerics, we calculate the anomalous scaling dimensions of the main cluster
growth, calling into question results much used in the literature. Apart from
the solution branch which originates from the exactly solvable case $\gamma =
1$, we find a new branch of solutions near $\gamma = 1/2$, which violates
scaling relations widely believed to hold universal.
