  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 $n^{2/3}(c_2+o(1))$ and that it converges to
the Tracy-Widom distribution after suitable scaling. Johansson further showed
that all maximal paths have a displacement of $n^{\frac23+o(1)}$ from the
diagonal with probability tending to one as $n\to \infty$. Here we prove that
the maximal length of an increasing path restricted to lie within a strip of
width $n^{\gamma}, \gamma<\frac23$, around the diagonal has expectation
$2n-n^{1-\gamma+o(1)}$, variance $n^{1 - \frac{\gamma}{2}+o(1)}$ and that it
converges to the Gaussian distribution after suitable scaling.
