zbMATH — the first resource for mathematics

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.

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