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We derive exact formulas for the expectation value of local observables in aone-dimensional gas of bosons with point-wise repulsive interactions(Lieb-Liniger model). Starting from a recently conjectured expression for theexpectation value of vertex operators in the sinh-Gordon field theory, wederive explicit analytic expressions for the one-point $K$-body correlationfunctions $\langle (\Psi^\dagger)^K(\Psi)^K\rangle$ in the Lieb-Liniger gas,for arbitrary integer $K$. These are valid for all excited states in thethermodynamic limit, including thermal states, generalized Gibbs ensembles andnon-equilibrium steady states arising in transport settings. Our formulasdisplay several physically interesting applications: most prominently, theyallow us to compute the full counting statistics for the particle-numberfluctuations in a short interval. Furthermore, combining our findings with therecently introduced generalized hydrodynamics, we are able to study multi-pointcorrelation functions at the Eulerian scale in non-homogeneous settings. Ourresults complement previous studies in the literature and provide a fullsolution to the problem of computing one-point functions in the Lieb Linigermodel.

From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics