Exact lower tail large deviations of the KPZ equation

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.