Longest increasing path within the critical strip

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 − n^1/3(c₁ + o(1)), variance n^2/3(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 n^γ, \(\gamma < {2 \over 3}\), around the diagonal has expectation 2n − n^1−γ+o(1), variance \({n^{1 - {\gamma \over 2} + o(1)}}\) and that it converges to the Gaussian distribution after suitable scaling.