  In the classic model of first passage percolation, for pairs of vertices
separated by a Euclidean distance $L$, geodesics exhibit deviations from their
mean length $L$ that are of order $L^\chi$, while the transversal fluctuations,
known as wandering, grow as $L^\xi$. We find that when weighting edges directly
with their Euclidean span in various spatial network models, we have two
distinct classes defined by different exponents $\xi=3/5$ and $\chi = 1/5$, or
$\xi=7/10$ and $\chi = 2/5$, depending only on coarse details of the specific
connectivity laws used. Also, the travel time fluctuations are Gaussian, rather
than Tracy-Widom, which is rarely seen in first passage models. The first class
contains proximity graphs such as the hard and soft random geometric graph, and
the $k$-nearest neighbour random geometric graphs, where via Monte Carlo
simulations we find $\xi=0.60\pm 0.01$ and $\chi = 0.20\pm 0.01$, showing a
theoretical minimal wandering. The second class contains graphs based on
excluded regions such as $\beta$-skeletons and the Delaunay triangulation and
are characterised by the values $\xi=0.70\pm 0.01$ and $\chi = 0.40\pm 0.01$,
with a nearly theoretically maximal wandering exponent. We also show
numerically that the KPZ relation $\chi = 2\xi -1$ is satisfied for all these
models. These results shed some light on the Euclidean first passage process,
but also raise some theoretical questions about the scaling laws and the
derivation of the exponent values, and also whether a model can be constructed
with maximal wandering, or non-Gaussian travel fluctuations, while embedded in
space.
