A universality property for last-passage percolation paths close to the axis

We consider a last-passage directed percolation model in Z₊², with i.i.d. weights whose common distribution has a finite (2+p)th moment. We study the fluctuctuations of the passage time from the origin to the point (n,[n^a]). We show that, for suitable a (depending on p), this quantity, appropriately scaled, converges in distribution as n → ∞ to the Tracy-Widom distribution, irrespective of the underlying weight distribution. The argument uses a coupling to a Brownian directed percolation problem and the strong approximation of Komlós, Major and Tusnédy.