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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 thefluctuations of the passage time from the origin to the point$\big(n,n^{\lfloor a \rfloor}\big)$. We show that, for suitable $a$ (dependingon $p$), this quantity, appropriately scaled, converges in distribution as$n\to\infty$ to the Tracy-Widom distribution, irrespective of the underlyingweight distribution. The argument uses a coupling to a Brownian directedpercolation problem and the strong approximation of Komlós, Major andTusnády

A Universality Property for Last-Passage Percolation Paths Close to the Axis