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 $$\{1,2,\dots,N\}$$. 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 $$N \geq 1$$ let $$\pi_N$$ be a uniformly chosen permutation with length $$N$$. Let $$l(\pi_N)$$ be the length of the longest, increasing subsequence of $$\pi_N$$.

Then we have for every $$x \in \mathbb{R}$$ that
 * $$\mathbb{P}\left(\frac{l(\pi_N)-2\sqrt{N}}{N^{1/6}}\leq x \right)\to F_2(x),\quad N \to \infty$$

where $$F_2(x)$$ is the Tracy-Widom distribution of the Gaussian unitary ensemble.