I think this should be equivalent to a directed, site first-passage percolation (FPP) model where each site takes a passage time of $2$ 1$or$\sqrt{2}$, each with probability$1/2$. I'm not convinced of this one (and haven't given it enough thought) but I'm pretty sure you can come up with a correspondence between some FPP model and this one. As R W says in another answer, Kingman's subadditive ergodic theorem implies that the passage time$a_n$between the origin and the point$(n,n)$should be of the order$\mu n$for some constant$\mu > 0$. As Ori Gurel-Gurevich says, the Benjamini-Kalai-Schramm estimate should apply so that the variance of the passage times is sublinear: $$a_n = \mu n + O(n/\log n).$$ Let me now state some conjectures: • The variance of the passage time should be$O(n^{2\chi})$, where$\chi = 1/3$. • Let$\gamma_n$denote the minimizing path from the origin to$(n,n)$. Let$d_n$be the maximum Euclidean distance that$\gamma_n$reaches away from the straight line path from the origin to$(n,n)$. Then$d_n$should be$O(n^\xi)$, for$\xi = 2/3$. • Similar statements should hold in arbitrary dimension, for different values of the constants$\chi$and$\xi$. Nonetheless, the constants should always obey the Kardar-Parisi-Zhang (KPZ) equation $$\chi = 2\xi - 1.$$ There has been much work on getting some precise bounds for these constants for FPP, but we are still far from rigorously proving that$\chi = 1/3$and$\xi = 2/3$. See the excellent survey Models of First-Passage Percolation by Howard (2004) for more on this topic. 1 This should be equivalent to a directed, site first-passage percolation (FPP) model where each site takes a passage time of$2$or$\sqrt{2}$, each with probability$1/2$. As R W says in another answer, Kingman's subadditive ergodic theorem implies that the passage time$a_n$between the origin and the point$(n,n)$should be of the order$\mu n$for some constant$\mu > 0$. As Ori Gurel-Gurevich says, the Benjamini-Kalai-Schramm estimate should apply so that the variance of the passage times is sublinear: $$a_n = \mu n + O(n/\log n).$$ Let me now state some conjectures: • The variance of the passage time should be$O(n^{2\chi})$, where$\chi = 1/3$. • Let$\gamma_n$denote the minimizing path from the origin to$(n,n)$. Let$d_n$be the maximum Euclidean distance that$\gamma_n$reaches away from the straight line path from the origin to$(n,n)$. Then$d_n$should be$O(n^\xi)$, for$\xi = 2/3$. • Similar statements should hold in arbitrary dimension, for different values of the constants$\chi$and$\xi$. Nonetheless, the constants should always obey the Kardar-Parisi-Zhang (KPZ) equation $$\chi = 2\xi - 1.$$ There has been much work on getting some precise bounds for these constants for FPP, but we are still far from rigorously proving that$\chi = 1/3$and$\xi = 2/3\$. See the excellent survey Models of First-Passage Percolation by Howard (2004) for more on this topic.