Hi,

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 of Seppalainen:

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.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 ?

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 ?