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We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equationin $1+1$ dimensions in curved (or droplet) geometry. We show that for shorttime $t$, the probability distribution $P(H,t)$ of the height $H$ at a givenpoint $x$ takes the scaling form $P(H,t) \sim \exp{\left(-\Phi_{\rmdrop}(H)/\sqrt{t} \right)}$ where the rate function $\Phi_{\rm drop}(H)$ iscomputed exactly. While it is Gaussian in the center, i.e., for small $H$, thePDF has highly asymmetric non-Gaussian tails which we characterize in detail.This function $\Phi_{\rm drop}(H)$ is surprisingly reminiscent of the largedeviation function describing the stationary fluctuations of finite size modelsbelonging to the KPZ universality class. Thanks to a recently discoveredconnection between KPZ and free fermions, our results have interestingimplications for the fluctuations of the rightmost fermion in a harmonic trapat high temperature and the full couting statistics at the edge.

Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature