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We derive exact analytic expressions for the $n$-body local correlations inthe one-dimensional Bose gas with contact repulsive interactions (Lieb-Linigermodel) in the thermodynamic limit. Our results are valid for arbitrary statesof the model, including ground and thermal states, stationary states after aquantum quench, and non-equilibrium steady states arising in transportsettings. Calculations for these states are explicitly presented and physicalconsequences are critically discussed. We also show that the $n$-body localcorrelations are directly related to the full counting statistics for theparticle-number fluctuations in a short interval, for which we provide anexplicit analytic result.

Exact local correlations and full counting statistics for arbitrary states of the one-dimensional interacting Bose gas