Take $n$ points uniformly in $[0,1] \times [0,1]$. Then pick uniformly $X_0$ one of these points as your starting point. Then let $X_1$ be the nearest neighbor of $X_0$, let $X_2$ be the nearest neighbor (not yet visited) of $X_1$ and so on. What can be said of the asymptotic of $X_n-X_{n-1}$ the length of the last crossed edge? What about the length of the longest crossed edge? I stumbled upon these kind of models in the area of environmental statistics where one tries to find clusters in a geographical dataset but I am not sure if this question is interesting.

Edit : Many more questions could be asked about this model : If one makes the model dynamic by adding the points in $[0,1]^2$ sequentially, most of the time the addition of an extra point changes the path only locally, but from time to time it will have a big impact. Does the path converges locally? (I guess not). How often do you see catastrophic modifications of the paths?

Finally, is there a way to find an interesting local spectrum in this object by renormalizing it and looking at the sizes of the edge ? It is probably related to the local dimension of the counting measure of the uniform PPP on [0,1]^2. But as pointed out in the answer below the first step would be to obtain the asymptotic of $L_n$ the total length.