  In this paper we continue our earlier work about topological first passage
percolation and answer certain questions asked in our previous paper. Notably,
we prove that apart from trivialities, in the generic configuration there
exists exactly one geodesic ray, in the sense that we consider two geodesic
rays distinct if they only share a finite number of edges. Moreover, we show
that in the generic configuration any not too small and not too large convex
set arises as the limit of a sequence $\frac{B(t_n)}{t_n}$ for some
$(t_n)\to\infty$. Finally, we define topological Hilbert first passage
percolation, and amongst others we prove that certain geometric properties of
the percolation in the generic configuration guarantee that we consider a
setting linearly isomorphic to the ordinary topological first passage
percolation.
