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This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

ADDENDUM: regarding fluctuations we have the paper of Benjamini, Kalai and Schramm. There's a newer paper by Benaïm and Rossignol, which I haven't read. I guess that most of the results apply to this setting as well.

This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

ADDENDUM: regarding fluctuations we have the paper of Benjamini, Kalai and Schramm, First Passage Percolation Has Sublinear Distance Variance. There's a newer paper by Benaïm and Rossignol, Exponential concentration for First Passage Percolation through modified Poincare inequalities, which I haven't read. I guess that most of the results apply to this setting as well.

This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

ADDENDUM: regarding fluctuations we have the paper of Benjamini, Kalai and Schramm. There's a newer paper by Benaïm and Rossignol, which I haven't read. I guess that most of the results apply to this setting as well.

This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

ADDENDUM: regarding fluctuations we have the paper of Benjamini, Kalai and Schramm. There's a newer paper by Benaïm and Rossignol, which I haven't read. I guess that most of the results apply to this setting as well.

This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

This is an interesting variation of First passage percolation (see this question). There's no reason to think that in this case the answer will be nice in any way, as percolation thresholds rarely are.

I think that the speed should be strictly larger than $\sqrt{2}$, for the following reason: if we had a path using NE diagonals almost all the time with high probability, then the graph consisting of only the long (say >100) streaks of NE diagonals should percolate. But standard percolation argument should show that this isn't the case. I hope this makes sense, I don't have time to check the details right now.

ADDENDUM: regarding fluctuations we have the paper of Benjamini, Kalai and Schramm. There's a newer paper by Benaïm and Rossignol, which I haven't read. I guess that most of the results apply to this setting as well.