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A universality property for last-passage percolation paths close to the axis. (English) Zbl 1111.60068
Summary: We consider a last-passage directed percolation model in \(\mathbb{Z}_+^2\), with i.i.d. weights whose common distribution has a finite \((2+p)\)th moment. We study the fluctuations 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 \(n\to\infty\) 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 Tusnády.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
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