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Transversal fluctuations for increasing subsequences on the plane. (English) Zbl 0960.60097

Summary: Consider a realization of a Poisson process in \(\mathbb{R}^2\) 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 \(\chi=1/3\) [see J. Baik, P. Deift and K. Johansson, J. Am. Math. Soc. 12, No. 4, 1119-1178 (1999; Zbl 0932.05001)]. 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 \(\chi=2\xi-1\) which is believed to hold in many random growth models.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Citations:

Zbl 0932.05001
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