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Not quite what Tom suggested, but this model is a form of last passage percolation (LPP).

There is always a minimal path that makes only N,E or NE steps. To see this, suppose a minimal path moves from $(i,j)$ to $(i-1,j+1)$, and that the next visit to column $i$ is at $(i,k)$. Using instead a straight segment from $(i,j)$ to $(i,k)$ cannot increase the path length. The other cases are similar.

It follows that to find the speed (or limit shape), we need to find the maximal number of NE shortcuts that can be used in a path to $(n,m)$, as each reduces the distance by $1$ (compared to the grid distance).

This is similar to LPP on $\{\mathbb Z}^2$ with Bernoulli weights (for which the limit shape is known), but is not the same since only one shortcut may be used in each row or column.

[edit: I had geometric weights in mind]

Not quite what Tom suggested, but this model is a form of last passage percolation (LPP).

There is always a minimal path that makes only N,E or NE steps. To see this, suppose a minimal path moves from $(i,j)$ to $(i-1,j+1)$, and that the next visit to column $i$ is at $(i,k)$. Using instead a straight segment from $(i,j)$ to $(i,k)$ cannot increase the path length. The other cases are similar.

It follows that to find the speed (or limit shape), we need to find the maximal number of NE shortcuts that can be used in a path to $(n,m)$, as each reduces the distance by $1$ (compared to the grid distance).

This is similar to LPP (Last Passage Percolation) on $\mathbb{Z}^2$ with Bernoulli weights (for which the limit shape is known), but is not the same since only one shortcut may be used in each row or column.

[edit: I had geometric weights in mind]

Not quite what Tom suggested, but this model is a form of last passage percolation (LPP).

There is always a minimal path that makes only N,E or NE steps. To see this, suppose a minimal path moves from $(i,j)$ to $(i-1,j+1)$, and that the next visit to column $i$ is at $(i,k)$. Using instead a straight segment from $(i,j)$ to $(i,k)$ cannot increase the path length. The other cases are similar.

It follows that to find the speed (or limit shape), we need to find the maximal number of NE shortcuts that can be used in a path to $(n,m)$, as each reduces the distance by $1$ (compared to the grid distance).

This is similar to LPP on $\{\mathbb Z}^2$ with Bernoulli weights (for which the limit shape is known), but is not the same since only one shortcut may be used in each row or column.

Not quite what Tom suggested, but this model is a form of last passage percolation (LPP).

There is always a minimal path that makes only N,E or NE steps. To see this, suppose a minimal path moves from $(i,j)$ to $(i-1,j+1)$, and that the next visit to column $i$ is at $(i,k)$. Using instead a straight segment from $(i,j)$ to $(i,k)$ cannot increase the path length. The other cases are similar.

It follows that to find the speed (or limit shape), we need to find the maximal number of NE shortcuts that can be used in a path to $(n,m)$, as each reduces the distance by $1$ (compared to the grid distance).

This is similar to LPP on $\{\mathbb Z}^2$ with Bernoulli weights (for which the limit shape is known), but is not the same since only one shortcut may be used in each row or column.

[edit: I had geometric weights in mind]

Not quite what Tom suggested, but this model is a form of last passage percolation.

There is always a minimal path that makes only N,E or NE steps. To see this, suppose a minimal path moves from $(i,j)$ to $(i-1,j+1)$, and that the next visit to column $i$ is at $(i,k)$. Using instead a straight segment from $(i,j)$ to $(i,k)$ cannot increase the path length. The other cases are similar.

It follows that to find the speed (or limit shape), we need to find the maximal number of NE shortcuts that can be used in a path to $(n,m)$, as each reduces the distance by $1$ (compared to the grid distance).

This is similar to LPP on $\{\mathbb Z}^2$ with Bernoulli weights (for which the limit shape is known), but is not the same since only one shortcut may be used in each row or column.

Not quite what Tom suggested, but this model is a form of last passage percolation (LPP).

There is always a minimal path that makes only N,E or NE steps. To see this, suppose a minimal path moves from $(i,j)$ to $(i-1,j+1)$, and that the next visit to column $i$ is at $(i,k)$. Using instead a straight segment from $(i,j)$ to $(i,k)$ cannot increase the path length. The other cases are similar.

It follows that to find the speed (or limit shape), we need to find the maximal number of NE shortcuts that can be used in a path to $(n,m)$, as each reduces the distance by $1$ (compared to the grid distance).

This is similar to LPP on $\{\mathbb Z}^2$ with Bernoulli weights (for which the limit shape is known), but is not the same since only one shortcut may be used in each row or column.