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

Understand

We present a systematic short time expansion for the generating function of the one point height probability distribution for the KPZ equation with droplet initial condition, which goes much beyond previous studies.

• The expansion is checked against a numerical evaluation of the known exact Fredholm determinant expression.
• We also obtain the next order term for the Brownian initial condition.
• Although initially devised for short time, a resummation of the series allows to obtain also the \textit{long time large deviation function}, found to agree with previous works using completely different techniques.