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We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ)equation, providing precise control over the left tail of the heightdistribution for narrow wedge initial condition. Our analysis exploits an exactconnection between the KPZ one-point distribution and the Airy point process --an infinite particle Coulomb-gas which arises at the spectral edge in randommatrix theory. We develop the large deviation principle for the Airy pointprocess and use it to compute, in a straight-forward and assumption-freemanner, the KPZ large deviation rate function in terms of an electrostaticproblem (whose solution we evaluate). This method also applies to thehalf-space KPZ equation, showing that its rate function is half of thefull-space rate function. In addition to these long-time estimates, we providerigorous proof of finite-time tail bounds on the KPZ distribution whichdemonstrate a crossover between exponential decay with exponent $3$ (in theshallow left tail) to exponent $5/2$ (in the deep left tail). The full-spaceKPZ rate function agrees with the one computed in Sasorov et al. [J. Stat.Mech, 063203 (2017)] via a WKB approximation analysis of a non-local,non-linear integro-differential equation generalizing Painlev\'e II which Amiret al. [Comm. Pure Appl. Math. 64, 466 (2011)] related to the KPZ one-pointdistribution.

Coulomb-gas electrostatics controls large fluctuations of the KPZ equation