If I have $n$ points uniformly distributed on the surface of a torus, and form a graph by adding an edge between pairs whenever they are within a unit distance (induced by the Euclidean metric), **I have a random geometric graph.**

**The Euclidean geodesic between two vertices** is just the shortest path, but where short means the Euclidean lengths of the edges along the path sum to the smallest value available (i.e. over all available paths).

The **passage time**, as well as **the maximum transversal wandering** of the path w.r.t. the straight line joining them, are conjectured to follow a power law $L^{\chi}$ and $L^{\xi}$, where $L$ is the Euclidean distance between the endpoints (norm of the straight line segment joining them). The first exponent $\chi$ gives the scaling of the root of the variance of the travel time, while the second $\xi$ gives instead the scaling of the expected wandering. See e.g. "The universal relation between scaling exponents in first-passage percolation", S. Chatterjee, Ann. Maths 177, 2013.

The relation $\chi = 2 \xi -1$ is meant to hold, relating the orders of the expected magnitude of the transversal wandering, and the time fluctuations about their expectation.

Though we don't know the exponents, **should such a relation also hold in the spatial setting of the random geometric graph**, where the edge lengths are not i.i.d., but induced by a point process?