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Fredholm determinants associated to deformations of the Airy kernel areclosely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equationwith narrow wedge initial data, and they also appear as largest particledistribution in models of positive-temperature free fermions. We show thatlogarithmic derivatives of the Fredholm determinants can be expressed in termsof a 2x2 Riemann-Hilbert problem, and we use this to derive asymptotics for theFredholm determinants. As an application of our result, we derive precise lowertail asymptotics for the solution of the KPZ equation with narrow wedge initialdata, refining recent results by Corwin and Ghosal.

A Riemann-Hilbert approach to the lower tail of the KPZ equation