  We study existence of percolation in the hierarchical group of order $N$,
which is an ultrametric space, and transience and recurrence of random walks on
the percolation clusters. The connection probability on the hierarchical group
for two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)},
\delta>-1$, with $c_k=C_0+C_1\log k+C_2k^\alpha$, non-negative constants $C_0,
C_1, C_2$, and $\alpha>0$. Percolation was proved in Dawson and Gorostiza
(2013) for $\delta<1$, and for the critical case, $\delta=1,C_2>0$, with
$\alpha>2$. In this paper we improve the result for the critical case by
showing percolation for $\alpha>0$. We use a renormalization method of the type
in the previous paper in a new way which is more intrinsic to the model. The
proof involves ultrametric random graphs (described in the Introduction). The
results for simple (nearest neighbour) random walks on the percolation clusters
are: in the case $\delta<1$ the walk is transient, and in the critical case
$\delta=1, C_2>0,\alpha>0$, there exists a critical $\alpha_c\in(0,\infty)$
such that the walk is recurrent for $\alpha<\alpha_c$ and transient for
$\alpha>\alpha_c$. The proofs involve graph diameters, path lengths, and
electric circuit theory. Some comparisons are made with behaviours of random
walks on long-range percolation clusters in the one-dimensional Euclidean
lattice.
