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  "body": "\n## Definition\n\nLet $\\mathrm{Ai}$ be the Airy function.\nLet $q \\colon \\R \\to \\R$ be the Hastings–McLeod solution of the Painlevé II equation\n$$q''(s) = s\\, q(s) + 2 q(s)^3,$$\nthat is, the unique solution with $q(s) \\sim \\mathrm{Ai}(s)$ as $s \\to +\\infty$.\nThe *Tracy–Widom distribution* (for $\\beta = 2$) is the [[probability distribution|0DVDCMA2DK]] on $\\R$ with distribution function\n$$F_2(s) = \\exp\\Big( -\\int_s^\\infty (x - s)\\, q(x)^2 \\, dx \\Big), \\qquad s \\in \\R.$$\nThe laws for $\\beta = 1$ and $\\beta = 4$ are given by\n$$F_1(s) = \\exp\\Big( -\\frac12 \\int_s^\\infty q(x)\\, dx \\Big) F_2(s)^{1/2}, \\qquad F_4\\Big(\\frac{s}{\\sqrt 2}\\Big) = \\cosh\\Big( \\frac12 \\int_s^\\infty q(x)\\, dx \\Big) F_2(s)^{1/2}.$$\n\n## Remarks\n\nExample: 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 $\\mathbb{E}\\abs{H_{ij}}^2 = 1$.\nIf $\\lambda_{\\max}$ is its largest [[eigenvalue|0BQAA2BK3Q]], then $n^{1/6} (\\lambda_{\\max} - 2\\sqrt n)$ [[converges in distribution|0SRYJGXKR4]] to $F_2$.\nEquivalently, $F_2(s) = \\det(I - K_{\\mathrm{Ai}})$ on $L^2(s, \\infty)$, the Fredholm determinant of the Airy kernel $K_{\\mathrm{Ai}}(x, y) = \\frac{\\mathrm{Ai}(x) \\mathrm{Ai}'(y) - \\mathrm{Ai}'(x) \\mathrm{Ai}(y)}{x - y}$.\nNon-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.\nBaik, 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.\n",
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