  We derive the leading asymptotic behavior and build a new series
representation for the Fredholm determinant of integrable integral operators
appearing in the representation of the time and distance dependent correlation
functions of integrable models described by a six-vertex R-matrix. This series
representation opens a systematic way for the computation of the long-time,
long-distance asymptotic expansion for the correlation functions of the
aforementioned integrable models away from their free fermion point. Our method
builds on a Riemann--Hilbert based analysis.
