This paper works out the large‑distance behaviour of a Fredholm determinant that appears when one studies dynamical two‑point correlation functions in the Lieb–Liniger Bose gas. The determinant belongs to an “integrable” integral operator whose kernel is a generalized version of the sine kernel. A parameter x controls rapid oscillations along the integration contour and plays the role of the distance variable in physical applications.

The authors set up the integral operator explicitly. Its kernel is written in the usual integrable form V(λ,µ) = E_L(λ)·E_R(µ)/(λ−µ), with the vector factors E_L and E_R built from four functional parameters u, g, ν and ϑ (theta). Those parameter functions are analytic in a horizontal strip of the complex plane: u, g and ν are holomorphic there, while ϑ is allowed to be meromorphic. The work replaces the determinant problem by a related matrix Riemann–Hilbert problem — a type of boundary‑value problem for matrices — and applies the Deift–Zhou nonlinear steepest‑descent method to extract the large‑x asymptotics.

Their main results are summarized in three theorems. The theorems give explicit formulas for the leading large‑x behaviour: the dominant exponential decay, the logarithmic corrections, and the constant term. All these pieces are written as functionals of the four parameter functions. The precise form of the constant and of subleading corrections depends on the number of poles that a certain combination of the parameter functions has on the real axis. The paper works out in detail the cases with no poles and with two poles, and it treats the general n‑pole situation with parts labelled as a static case and a conjecture.

Why this matters: the Fredholm determinant studied here is an auxiliary “generating” object that appears when one expands dynamical correlation functions into form‑factor series at finite coupling. Knowing its large‑distance asymptotics is the key input for obtaining the long‑distance, long‑time behaviour of correlation functions in the Lieb–Liniger model at finite temperature and for different equilibrium states, including generalized Gibbs ensembles (a statistical state that keeps several conserved quantities). The analysis is carried out with full mathematical control for the model problem, so it offers a rigorous tool that can be adapted to physical applications.