Mathematics

Dr. Mathew Joseph
University of Sheffield

What can we say about the longest increasing subsequence in a random uniform permutation? This seemingly simple sounding question, first posed by Ulam, has occupied mathematicians for over half a century now and has led to beautiful connections between analysis, probability and representation theory. An almost equivalent reformulation of the problem is the following. Consider the square $[0,n]^2$ with points from a Poisson point process of intensity 1 distributed within it. What is the behaviour of the number of points $L_n$ (length) on a maximal increasing path (an increasing path that contains the most number of points)? In a seminal work, Baik, Deift and Johansson proved that when properly centered and scaled, $L_n$ converges to the mysterious Tracy-Widom distribution, which first appeared in random matrix theory. Later Johansson showed that all maximal paths lie within the strip of width $n^{\\frac{2}{3} +o(1)}$ around the diagonal with probability tending to 1 as $n \\to \\infty$. We shall discuss recent work on the Gaussian behaviour of the length $L_n^{(\\gamma)}$ of a maximal increasing path restricted to lie within a strip of width $n^{\\gamma}, \\gamma< \\frac{2}{3}$. This is based on joint work with Partha Dey and Ron Peled.