# Continuum limit of first-passage percolation paths

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.

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If that is last passage percolation instead, such a limit is related to directed polymers in random media and with KPZ and stochastic heat equations. Maybe a google scholar search would be helpful here. –  Leonid Petrov Mar 19 '13 at 17:51
@Leonid Petrov: Thank you for your comment. I did have a look at the literature, but the problem I'm interested it doesn't seem to have received a lot of attention in the past. –  Eckhard Mar 19 '13 at 18:29