Tracy–Widom distribution

Remarks

Example: let $H_n$ be an $n \times n$ matrix of the Gaussian unitary ensemble, Hermitian with independent standard real normal diagonal entries and independent complex normal entries above the diagonal with $\mathbb{E}\abs{H_{ij}}^2 = 1$. If $\lambda_{\max}$ is its largest eigenvalue, then $n^{1/6} (\lambda_{\max} - 2\sqrt n)$ converges in distribution to $F_2$. Equivalently, $F_2(s) = \det(I - K_{\mathrm{Ai}})$ on $L^2(s, \infty)$, the Fredholm determinant of the Airy kernel $K_{\mathrm{Ai}}(x, y) = \frac{\mathrm{Ai}(x) \mathrm{Ai}'(y) - \mathrm{Ai}'(x) \mathrm{Ai}(y)}{x - y}$. Non-example: the standard normal law is not a Tracy–Widom law; $F_2$ has mean about $-1.77$ and is not symmetric about its mean. Baik, Deift and Johansson showed that the length $\ell_n$ of the longest increasing subsequence of a uniform random permutation of $n$ letters satisfies $(\ell_n - 2\sqrt n)/n^{1/6} \to F_2$ in distribution.