Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature

We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability distribution P (H, t) of the height H at a given point x takes the scaling form P (H, t) ∼exp-Φ₃ₑ₎₏ (H) /sqrt[t] where the rate function Φ₃ₑ₎₏ (H) is computed exactly for all H. While it is Gaussian in the center, i. e. , for small H, the probability distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ₃ₑ₎₏ (H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.