  We study percolation on the hierarchical lattice of order $N$ where the
probability of connection between two points separated by distance $k$ is of
the form $c_k/N^{k(1+\delta)},\; \delta >-1$. Since the distance is an
ultrametric, there are significant differences with percolation on the
Euclidean lattice. There are two non-critical regimes: $\delta <1$, where
percolation occurs, and $\delta >1$, where it does not occur. In the critical
case, $\delta =1$, we use an approach in the spirit of the renormalization
group method of statistical physics and connectivity results of Erd\H{o}s-Renyi
random graphs play a key role. We find sufficient conditions on $c_k$ such that
percolation occurs, or that it does not occur. An intermediate situation called
pre-percolation is also considered. In the cases of percolation we prove
uniqueness of the constructed percolation clusters. In a previous paper
\cite{DG1} we studied percolation in the $N\to\infty$ limit (mean field
percolation) which provided a simplification that allowed finding a necessary
and sufficient condition for percolation. For fixed $N$ there are open
questions, in particular regarding the existence of a critical value of a
parameter in the definition of $c_k$, and if it exists, what would be the
behaviour at the critical point.
