TY - JOUR

T1 - Longest increasing path within the critical strip

AU - Dey, Partha S.

AU - Joseph, Mathew

AU - Peled, Ron

N1 - Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.

PY - 2023

Y1 - 2023

N2 - A Poisson point process of unit intensity is placed in the square [0, n]2. An increasing path is a curve connecting (0, 0) with (n, n) which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process which it passes through. Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation 2n − n 1/3(c 1 + o(1)), variance n 2/3(c 2 + o(1)) for some c 1, c 2 > 0 and that it converges to the Tracy–Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of n23+o(1) from the diagonal with probability tending to one as n → ∞. Here we prove that the maximal length of an increasing path restricted to lie within a strip of width n γ, γ<23 , around the diagonal has expectation 2n − n 1−γ+o(1), variance n1−γ2+o(1) and that it converges to the Gaussian distribution after suitable scaling.

AB - A Poisson point process of unit intensity is placed in the square [0, n]2. An increasing path is a curve connecting (0, 0) with (n, n) which is non-decreasing in each coordinate. Its length is the number of points of the Poisson process which it passes through. Baik, Deift and Johansson proved that the maximal length of an increasing path has expectation 2n − n 1/3(c 1 + o(1)), variance n 2/3(c 2 + o(1)) for some c 1, c 2 > 0 and that it converges to the Tracy–Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of n23+o(1) from the diagonal with probability tending to one as n → ∞. Here we prove that the maximal length of an increasing path restricted to lie within a strip of width n γ, γ<23 , around the diagonal has expectation 2n − n 1−γ+o(1), variance n1−γ2+o(1) and that it converges to the Gaussian distribution after suitable scaling.

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U2 - 10.1007/s11856-023-2603-8

DO - 10.1007/s11856-023-2603-8

M3 - Article

AN - SCOPUS:85180188681

SN - 0021-2172

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -