Asymptotics of a cubic sine kernel determinant

TL;DR

This paper analyzes the asymptotic behavior of a cubic sine kernel determinant, which generalizes the sine kernel in random matrix theory and quantum physics, using Riemann-Hilbert techniques for large interval sizes.

Contribution

It provides the first large-s asymptotics for a cubic sine kernel determinant for all real parameter values, extending understanding of integrable operators in physics and mathematics.

Findings

Derived explicit large s asymptotics for the determinant

Unified analysis for all real gamma values

Connects kernel asymptotics with physical models

Abstract

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