Transversal fluctuations for increasing subsequences on the plane

Consider a realization of a Poisson process in ℝ² with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N ^χ, where χ = 1/3, BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order N ^ξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1 which is believed to hold in many random growth models. Received: 16 April 1999 / Revised version: 5 July 1999 / Published online: 14 February 2000