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

Abstract

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 fluctuations of the passage time from the origin to the point $(n,n^{\lfloor a\rfloor })$. 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.

8 pages

References in corpus (2)

• Distribution functions for largest eigenvalues and their applications
• Limiting shape for directed percolation models