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