The Tracy-Widom Distribution at Large Dyson Index

We study the Tracy-Widom (TW) distribution fβ(a) in the limit of large Dyson index β→+∞. This distribution describes the fluctuations of the rescaled largest eigenvalue a1 of the Gaussian (alias Hermite) ensemble (GβE) of (infinitely) large random matrices. We show that, at large β, its probability density function takes the large deviation form fβ(a)∼e-βΦ(a). While the typical deviation of a1 around its mean is Gaussian of variance O(1/β), 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 Φ(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 Φ(a) numerically for all a and compare with exact numerical computations of the TW distribution at finite β. 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 a1>a2>⋯ which describes all edge eigenvalues of the GβE, and correspond to (minus) the higher energy levels of the SAO, obtaining large deviation forms for the marginal distribution of ai, the joint distributions, and the gap distributions.