A few years ago, when I was working on first-passage percolation problems, I thought about the following problem. Recently it came back to my mind.


Consider, for some $\delta=n^{-1}>0$, the grid $\delta\mathbb{Z} \times \delta\mathbb{Z}$ and associate with each edge $e$ an independent random weight $w(e)$. The edge weights may be assumed to be positive and drawn from a continuous distribution. Then, with probability one, there exists a unique path $p^\delta$ from, say, $(0,0)$ to $(1,0)$ that minimises the sum of edge weights along the path. The distribution of $p^\delta$ defines a measure $\mu^\delta$ on the space of non-intersecting paths in $\delta\mathbb{Z} \times \delta\mathbb{Z}$ from $(0,0)$ to $(0,1)$.


**Question:** Is anything known about $\lim\_{\delta\to 0}\mu^\delta$?


Intuitively, if the limit exists, it could be interpreted as a measure on the space of continuous curves in $\mathbb{Z}^2$ from $(0,0)$ to $(1,0)$. I am interested in any references, partial/conjectural results, or simulations.



Continuum limit of first-passage percolation paths
Due to what's written in the text of Benjamini, I would expect it to converge to a straight line, (unless, perhaps, if your law has too heavy log-tails, I'm not sure): see
<http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf>, section 2: *«Large balls converge after rescaling to a convex
centrally symmetric shape...»*


(If this shape in strictly convex, what seems to me more or less natural to expect, then the geodesic for the limit metric is unique, and is the straight line.)


