Asymptotics of a cubic sine kernel determinant

Abstract

We study the one parameter family of Fredholm determinants $\det (I-\gamma K_{\text{csin}}),\gamma \in \mathbb{R}$ of an integrable Fredholm operator $K_{\text{csin}}$ 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 $\det (I-\gamma K_{\text{csin}})$ for all values of the real parameter $\gamma$.

59 pages, 10 figures. arXiv admin note: text overlap with arXiv:1209.5415