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Asymptotics of a cubic sine kernel determinant

We study the one parameter family of Fredholm determinants $\det(I-γK_{\textnormal{csin}}),γ\in\mathbb{R}$ of an integrable Fredholm operator $K_{\textnormal{csin}}$ acting on the interval $(-s,s)$ whose kernel is a cubic generalization of the sine kernel which appears in random matrix theory. This Fredholm determinant appears in the description of the Fermi distribution of semiclassical non-equilibrium Fermi states in condensed matter physics as well as in random matrix theory. Using the Riemann-Hilbert method, we calculate the large $s$-asymptotics of $\det(I-γK_{\textnormal{csin}})$ for all values of the real parameter $γ$.