  Consider a realization of a Poisson process in R^2 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^chi,
where chi=1/3 by a result of Baik-Deift-Johansson. Here we show that typical
deviations of a maximal path from the diagonal x=y is of order N^xi with
xi=2/3. This is consistent with the scaling identity chi=2xi-1, which is
believed to hold in many random growth models.
