TITLE: Shortest grid-graph paths with random diagonal shortcuts
QUESTION [24 upvotes]: Suppose you have a network of edges connecting
each integer lattice point
in the 2D square grid $[0,n]^2$
to each of its (at most) four neighbors, {N,S,E,W}.
Within each of the $n^2$ unit cells of this grid,
select one of the two diagonals at random to
add to the network.
These diagonals serve as local "short cuts."
One could ask many questions about this model,
but let me start with this one:

What is the expected length of the shortest path
  in such a network from $(0,0)$ to $(n,n)$?

Here is an example for $n=25$:

      

Here a shortest path has length
$10+20 \sqrt{2} \approx 38.3$
in comparison to the shortest possible length,
$25 \sqrt{2} \approx 35.4$.
For small $n$, the growth rate of the length of the shortest
path appears to be linear in $n$, with a slope of about
1.52315.

      

I would appreciate learning if anyone recognizes
this model and/or knows the true growth rate. Thanks!
Edit. One more figure to address a question raised by jc.
This shows 10 shortest paths for $n=50$, for different random diagonal
choices (not shown).  But be aware that usually several shortest paths are tied as equally long,
and my code selects one with a systematic bias toward the lower right corner.
But the figure provides a sense of the variation.

REPLY [7 votes]: I also think, as already said, that this is exactly a problem of First Passage Percolation. I precise Tom's idea: take the whole lattice $\mathbb{Z}^2$ and add S0-NE diagonals (this is in fact a triangular lattice). Then put deterministic edge-weights equal to $1$ on the horizontal and vertical edges, and put independently on each diagonal edge a weight equal to $\sqrt{2}$ with probability $1/2$ and to $2$ (or anything larger than $2$) with probability one half. I think this is exactly equivalent to your model, no ?
The edge-weights are not i.i.d, but they are independent and stationnary (and thus ergodic). Kingman's result applies to show the existence of an almost sure limit.
Concerning fluctuations, Benjamini, Kalai and Schramm's argument should apply as is (it works on any lattice), but this should not be optimal as mentioned by Tom.
For the possibility to compute the value of the limit, maybe take a look at this article:

Timo Seppäläinen, Exact limiting shape for a simplified model of first-passage percolation on the plane, Ann. Probab. 26 (3) (1998) 1232–1250, https://doi.org/10.1214/aop/1022855751

Anyway, this is a nice model !
@ Omer: You said "LPP on $\mathbb{Z}^2$ with Bernoulli weights (for which the limit shape is known)" ... are you sure the shape is known ?