Transversal fluctuations for increasing subsequences on the plane
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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 Nχ, where χ = 1/3, [BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order Nξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1 which is believed to hold in many random growth models.
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