Transversal fluctuations for increasing subsequences on the plane

1999/10/27 by Kurt Johansson, Johansson, Kurt · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60K35 #82C24 #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60K35 #msc:82C24

paper · pdf · doi:10.48550/arxiv.math/9910146

16 pages

arxiv created 1999/10/27 · openalex publication_date 1999/10/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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^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.

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