# Transversal deviation in first passage percolation

Take the lattice $$\mathbb{L}^{2}=(\mathbb{Z}^{2},\mathbb{E}^{2})$$ with i.i.d. $$\text{U}[0,1]$$ weights on the edges, and the random variable $$D$$ giving the maximal transversal deviation of the geodesic path between two points $$\mathbf{x},\mathbf{y} \in \mathbb{Z}^{2}$$. This is unique, and formed from the minima over all edge-weighted paths between the points.

An example on a proximity graph on random points is shown. The vertical line is the random variable giving the transversal deviation.

What is known about the distribution of $$D$$?

$$T$$, the length of the paths, follows the Tracy-Widom distribution in the case where the weights are i.i.d., at least in some settings e.g. on $$\mathbb{L}^{2}$$.

For example, I have this for the case of a the Delaunay triangulation on random points on a torus, where the weights are now the lengths of the edges in Euclidean space: