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Longest increasing path within the critical strip

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Abstract

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 − n1/3(c1 + o(1)), variance n2/3(c2 + o(1)) for some c1, c2 > 0 and that it converges to the Tracy–Widom distribution after suitable scaling. Johansson further showed that all maximal paths have a displacement of \({n^{{2 \over 3} + o(1)}}\) 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 nγ, \(\gamma < {2 \over 3}\), around the diagonal has expectation 2n − n1−γ+o(1), variance \({n^{1 - {\gamma \over 2} + o(1)}}\) and that it converges to the Gaussian distribution after suitable scaling.

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Acknowledgements

We thank the referee for their comments which improved the exposition of the paper. We thank Lucas Journel for a careful reading of an earlier draft and for suggesting several improvements. We thank Eitan Bachmat for interesting discussions of related problems and application areas. Most of this work was completed while M.J. was at the University of Sheffield, and he thanks the School of Mathematics and Statistics for a supportive environment. The work of R.P. was supported in part by Israel Science Foundation grant 861/15 and the European Research Council starting grant 678520 (LocalOrder).

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St, Urbanas, IL, 61801, USA

    Partha S. Dey

  2. Indian Statistical Institute Bangalore, 8th Mile Mysore Road, RVCE Post, Bengaluru, 560059, India

    Mathew Joseph

  3. School of Mathematical Sciences, Tel Aviv University, 231 Schreiber Building, Ramat Aviv, Tel Aviv, 69978, Israel

    Ron Peled

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  1. Partha S. Dey
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Correspondence to Mathew Joseph.

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Dey, P.S., Joseph, M. & Peled, R. Longest increasing path within the critical strip. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2603-8

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