Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential

We present a complete theory for the full particle statistics of the positions of bulk and extremal particles in a one-dimensional Coulomb Gas (CG) with an arbitrary potential, in the typical and large deviations regimes. Typical fluctuations are described by a universal function which depends solely on general properties of the external potential. The rate function controlling large deviations is, rather unexpectedly, not strictly convex and has a discontinuous third derivative around its minimum for both extremal and bulk particles. This implies, in turn, that the rate function cannot predict the anomalous scaling of the typical fluctuations with the system size. Moreover, its asymptotic behavior for extremal particles differs from the predictions of the Tracy-Widom distribution. Thus many of the paradigmatic properties of the full particle statistics of two-dimensional systems do \emph{not} carry out into their one-dimensional counterparts, hence proving that 1d CG belongs to a different universality class. Our analytical expressions are thoroughly compared with Monte Carlo simulations showing excellent agreement.