Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Consider the following first-passage percolation problem on the plane grid $\mathbb{Z}^2$: all the horizontal edges are directed (pointing east) and carry an independent random weight, say standard exponential. All the vertical edges are undirected and carry a deterministic weight, say $c$.

As usual, the length of a path between two vertices is the sum of the weights of its edges. I am interested in the geometry of the shortest path between the origin and the point $(n,0)$. I am particular interested in the limit $n\to\infty$ when $c=c_n$ is allowed to vary (get smaller) with $n$

Question: Has this model been studied before? What are recommended references to start reading about it?

Below is a picture for $n=100$ and $c=0.05$;

enter image description here

Edit: The specification that the horizontal edges are directed is superfluous. Having undirected horizontal edges results in the same model.

share|improve this question
2  
Nice and natural model! –  Gil Kalai Jan 19 at 17:08
    
Simulations suggest that the fluctuations of the length of the shortest path are of order $n^{1/3}$, as is expected for an FPP model. –  Eckhard Feb 1 at 18:21

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.