We study the Tracy-Widom (TW) distribution \(f_\beta (a)\) in the limit of large Dyson index \(\beta \rightarrow +\infty \) . This distribution describes the fluctuations of the rescaled largest eigenvalue \(a_1\) of the Gaussian (alias Hermite) ensemble (G \(\beta \) E) of (infinitely) large random matrices. We show that, at large \(\beta \) , its probability density function takes the large deviation form \(f_\beta (a) \sim e^{-\beta \Phi (a)}\) . While the typical deviation of \(a_1\) around its mean is Gaussian of variance \(O(1/\beta )\) , this large deviation form describes the probability of rare events with deviation O(1), and governs the behavior of the higher cumulants. We obtain the rate function \(\Phi (a)\) as a solution of a Painlevé II equation. We derive explicit formula for its large argument behavior, and for the lowest cumulants, up to order 4. We compute \(\Phi (a)\) numerically for all a and compare with exact numerical computations of the TW distribution at finite \(\beta \) . These results are obtained by applying saddle-point approximations to an associated problem of energy levels \(E=-a\) , for a random quantum Hamiltonian defined by the stochastic Airy operator (SAO). We employ two complementary approaches: (i) we use the optimal fluctuation method to find the most likely realization of the noise in the SAO, conditioned on its ground-state energy being E (ii) we apply the weak-noise theory to the representation of the TW distribution in terms of a Ricatti diffusion process associated to the SAO. We extend our results to the full Airy point process \(a_1>a_2>\dots \) which describes all edge eigenvalues of the G \(\beta \) E, and correspond to (minus) the higher energy levels of the SAO, obtaining large deviation forms for the marginal distribution of \(a_i\) , the joint distributions, and the gap distributions.
