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

TL;DR

This paper develops a systematic short time expansion for the KPZ equation's height distribution, extends it to long times, and applies it to linear statistics of the GUE edge and trapped fermions, revealing new large deviation functions and corrections.

Contribution

It introduces a novel short time expansion method for the KPZ equation and applies it to GUE edge eigenvalues and trapped fermions, providing new analytical results and corrections.

Findings

Short time expansion matches numerical evaluations.

Derived long time large deviation functions.

Obtained higher-order cumulants and corrections for the Airy point process.

Abstract

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. Unexpected similarities with stationary large deviations of TASEP with periodic and open boundaries are discussed. Two additional applications are given. (i) Our method is generalized to study the linear statistics of the {Airy point process}, i.e. of the GUE edge eigenvalues. We obtain the generating function of the cumulants of the empirical measure to a high order. The second cumulant is found to match the result in the bulk obtained from the Gaussian free field by Borodin and Ferrari, but we obtain systematic corrections to the Gaussian free field (higher cumulants, expansion towards the edge). This also extends a result of Basor and Widom to a much higher order. We obtain {large deviation functions} for the {Airy point process} for a variety of linear statistics test functions. (ii) We obtain results for the \textit{counting statistics of trapped fermions} at the edge of the Fermi gas in both the high and the low temperature limits.