The Tracy-Widom distribution at large Dyson index

We study the Tracy-Widom (TW) distribution $f_\beta(a)$ in the limit of large Dyson index $\beta \to +\infty$. 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 Painlev\'{e} 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.