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 ${\displaystyle \{1,2,\dots ,N\}}${\displaystyle \{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.^{[1] [2]}

For each ${\displaystyle N\geq 1}${\displaystyle N\geq 1} let ${\displaystyle \pi _{N}}${\displaystyle \pi _{N}} be a uniformly chosen permutation with length ${\displaystyle N}${\displaystyle N}. Let ${\displaystyle l(\pi _{N})}${\displaystyle l(\pi _{N})} be the length of the longest, increasing subsequence of ${\displaystyle \pi _{N}}${\displaystyle \pi _{N}}.

Then we have for every ${\displaystyle x\in \mathbb {R} }${\displaystyle x\in \mathbb {R} } that

${\displaystyle \mathbb {P} \left({\frac {l(\pi _{N})-2{\sqrt {N}}}{N^{1/6}}}\leq x\right)\to F_{2}(x),\quad N\to \infty }${\displaystyle \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 ${\displaystyle F_{2}(x)}${\displaystyle F_{2}(x)} is the Tracy-Widom distribution of the Gaussian unitary ensemble.

• 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 .

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