Correlations in non-equilibrium Luttinger liquid and singular Fredholm determinants

We study interaction-induced correlations in Luttinger liquid with multiple Fermi edges. Many-particle correlation functions are expressed in terms of Fredholm determinants $\mathrm{d}\mathrm{e}\mathrm{t}(1+\widehat{A}\widehat{B})$, where $A(ϵ)$ and $B(t)$ have multiple discontinuities in energy and time spaces. Such determinants are a generalization of Toeplitz determinants with Fisher-Hartwig singularities. We propose a general asymptotic formula for this class of determinants and provide analytical and numerical support to this conjecture. This allows us to establish non-equilibrium power-law singularities of many-particle correlation functions. As an example, we calculate a two-particle distribution function characterizing correlations between left- and right-moving fermions that have left the interaction region.