Longest Increasing Path within the Critical Strip

Consider the square [0, n] with points from a Poisson point process of intensity 1 distributed within it. In [4] it was shown that the length Ln of a maximal increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later [16] showed that all maximal paths have a displacement of n 2 3 +o(1) from the diagonal with probability tending to 1 as n → ∞. In this paper, we consider the length L (γ) n of maximal increasing paths restricted to lie within a strip of width n γ , γ < 2 3 around the diagonal and show that when properly centered and scaled it converges to a Gaussian distribution. We also obtain bounds on the expectation and variance of L (γ) n .