  In this paper, we establish the first large deviation bounds for the Airy
point process. The proof is based on a novel approach which relies upon the
approximation of the Airy point process using Gaussian unitary ensemble (GUE)
up to exponentially small probability, together with precise estimates for the
stochastic Airy operator and edge rigidity for beta ensembles.
  As a by-product of our estimates for the Airy point process, we significantly
improve upon previous results on the lower tail probability of the one-point
distribution of the KPZ equation with narrow-wedge initial data and the
half-space KPZ equation with Neumann boundary parameter $A=-\frac{1}{2}$ in a
unified and much shorter manner. Our bounds hold for all sufficiently large
time $T$, and for the first time establish sharp super-exponential decay with
exponent $3$ for tail depth less than $T^{\frac{2}{3}}$ (with sharp leading
pre-factors $\frac{1}{12}$ and $\frac{1}{24}$ for tail depth less than $T^{
\frac{1}{6}}$).
