  Consider the Hopf--Cole solution $ h(t,x) $ of the KPZ equation with narrow
wedge initial condition. Regarding $ t\to\infty $ as a scaling parameter, we
provide the first rigorous proof of the Large Deviation Principle (LDP) for the
lower tail of $ h(2t,0)+\frac{t}{12} $, with speed $ t^2 $ and an explicit rate
function $ \Phi_-(z) $. This result confirms existing physic predictions
[Sasorov, Meerson, Prolhac 17], [Corwin, Ghosal, Krajenbrink, Le Doussal, Tsai
18], and [Krajenbrink, Le Doussal, Prolhac 18]. Our analysis utilizes the
formula from [Borodin, Gorin 16] to convert LDP of the KPZ equation to
calculating an exponential moment of the Airy point process. To estimate this
exponential moment, we invoke the stochastic Airy operator, and use the Riccati
transform, comparison techniques, and certain variational characterizations of
the relevant functional.
