Tracy–Widom distribution

Summary

The Tracy–Widom distributions are the limit laws of the largest eigenvalue of large random Hermitian or symmetric matrices, after centring and scaling. There are three of them, $F_1$, $F_2$ and $F_4$, for the orthogonal, unitary and symplectic Gaussian ensembles. They are given by a solution of the Painlevé II equation, or by a Fredholm determinant with the Airy kernel. They also appear far from matrices, for example for the longest increasing subsequence of a random permutation and for random growth models.