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We present a systematic short time expansion for the generating function ofthe one point height probability distribution for the KPZ equation with dropletinitial condition, which goes much beyond previous studies. The expansion ischecked against a numerical evaluation of the known exact Fredholm determinantexpression. We also obtain the next order term for the Brownian initialcondition. Although initially devised for short time, a resummation of theseries allows to obtain also the \textit{long time large deviation function},found to agree with previous works using completely different techniques.Unexpected similarities with stationary large deviations of TASEP with periodicand open boundaries are discussed. Two additional applications are given. (i)Our method is generalized to study the linear statistics of the {Airy pointprocess}, i.e. of the GUE edge eigenvalues. We obtain the generating functionof the cumulants of the empirical measure to a high order. The second cumulantis found to match the result in the bulk obtained from the Gaussian free fieldby Borodin and Ferrari, but we obtain systematic corrections to the Gaussianfree field (higher cumulants, expansion towards the edge). This also extends aresult of Basor and Widom to a much higher order. We obtain {large deviationfunctions} for the {Airy point process} for a variety of linear statistics testfunctions. (ii) We obtain results for the \textit{counting statistics oftrapped fermions} at the edge of the Fermi gas in both the high and the lowtemperature limits.

Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions