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.

Literature

• Romik, Dan (2015). The Surprising Mathematics of Longest Increasing Subsequences. doi:10.1017/CBO9781139872003. ISBN 9781107075832.
• Corwin, Ivan (2018). "Commentary on "Longest increasing subsequences: From patience sorting to the Baik–Deift–Johansson theorem" by David Aldous and Persi Diaconis". Bulletin of the American Mathematical Society. 55 (3): 363–374. doi:10.1090/bull/1623.