Critical Case
The critical case is where . Intuitively this is because the first-passage metric has no preference over the lengths of the edges hence there are many more paths to a specific vertex
of optimal length. This isn't the case when
since for
we have that
is concave which means that longer edges are preferred therefore the only path to
which can be made in optimal time is the direct edge from the origin. Similarly, when
we have that
is convex and hence shorter edges are preferred. This means that the only paths to
which can be of optimal length consist of
edges of length
.
In this case our simulations suggested that the limiting shape of should coincide precisely with the unit
ball
. The figure below shows one quadrant from a simulation where the edge weights have distribution
for . This is a distribution with an atom at
and an exponential tail. The white region shows the unoccupied space, the black region shows the active vertices under the LRFPP model and the grey points are those which are active in both the nearest neighbour and long-range models. For this simulation the nearest neighbour model has been coupled with the long range one so that the nearest neighbour growth set cannot extend further than the corresponding long range one. The growth set for the long-range model has a flat piece which extends over the entire edge of the
ball and the nearest neighbour growth set has a flat piece which coincides with the results of Marchand.
Our main result for the critical case is indeed exactly what the simulations have suggested in that there are only finitely many lattice points which are not reached in optimal time.
Theorem 1:
If we have that for
that the limiting shape:
exists a.s. and