  We investigate a probabilistic model for routeing in a multihop ad-hoc
communication network, where each user sends a message to the base station.
Messages travel in hops via other users, used as relays. Their trajectories are
chosen at random according to a Gibbs distribution, which favours trajectories
with low interference, measured in terms of signal-to-interference ratio. This
model was introduced in our earlier paper [KT18], where we expressed, in the
limit of a high density of users, the typical distribution of the family of
trajectories in terms of a law of large numbers. In the present work, we derive
its qualitative properties. We analytically identify the emerging typical
scenarios in three extreme regimes. We analyse the typical number of hops and
the typical length of a hop, and the deviation of the trajectory from the
straight line, (1) in the limit of a large communication area and large
distances, and (2) in the limit of a strong interference weight. In both
regimes, the typical trajectory approaches a straight line quickly, in regime
(1) with equal hop lengths. Interestingly, in regime (1), the typical length of
a hop diverges logarithmically in the distance of the transmitter to the base
station. We further analyse (3) local and global repulsive effects of a densely
populated subarea on the trajectories.
