Probability seminar: Longest increasing path within the critical strip
Mathew Joseph, University of Sheffield
SM3, School of Mathematics
Mathew Joseph, Sheffield
Longest increasing path within the critical strip
Consider the square [0, n]2 with points from a Poisson point process of intensity 1 distributed within it. In a seminal work, Baik, Deift and Johansson proved that the number of points Ln (length) on 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 Johansson showed that all maximal paths lie within the strip of width n2/3+ε around the diagonal with probability tending to 1 as n → ∞. We shall discuss recent work on the Gaussian behaviour of the length Ln(γ) of a maximal increasing path restricted to lie within a strip of width nγ, γ<2/3. This is based on joint work with Partha Dey and Ron Peled.