Baik–Deift–Johansson theorem

The Baik–Deift–Johansson theorem is a result from probabilistic combinatorics. It deals with the subsequences of a randomly uniformly drawn permutation from the set. The theorem makes a statement about the distribution of the length of the longest increasing subsequence in the limit. The theorem was influential in probability theory since it connected the KPZ-universality with the theory of random matrices.
The theorem was proven in 1999 by Jinho Baik, Percy Deift and Kurt Johansson.

Statement

For each let be a uniformly chosen permutation with length. Let be the length of the longest, increasing subsequence of.
Then we have for every that
where is the Tracy-Widom distribution of the Gaussian unitary ensemble.