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.

单位强度的泊松点过程被放置在正方形 [0, n ] ²中。递增路径是连接 (0, 0) 和 ( n, n ) 的曲线，该曲线在每个坐标上都是非递减的。它的长度是它经过的泊松过程的点数。Baik、Deift 和 Johansson 证明，对于某些c ，递增路径的最大长度具有期望 2 n − n ^1/3 ( c ₁ + o (1))，方差n ^2/3 ( c ₂ + o (1)) ₁ , c ₂ > 0 并且在适当的缩放后它收敛到 Tracy-Widom 分布。Johansson 进一步表明，所有最大路径相对于对角线的位移为\({n^{{2 \over 3} + o(1)}}\) ，且当n → ∞时概率趋于 1 。这里我们证明，一条递增路径的最大长度限制在宽度为n ^γ的条带内，\(\gamma < {2 \over 3}\)，围绕对角线的期望为 2 n − n ^{1− γ + o (1)}，方差\({n^{1 - {\gamma \over 2} + o(1)}}\)并且在适当的缩放后收敛到高斯分布。