Gap probability and full counting statistics in the one dimensional one-component plasma

We consider the $1d$ one-component plasma (OCP) in thermal equilibrium, consisting of $N$ equally charged particles on a line, with pairwise Coulomb repulsion and confined by an external harmonic potential. We study two observables: (i) the distribution of the gap between two consecutive particles in the bulk and (ii) the distribution of the number of particles $N_I$ in a fixed interval $I=[-L,+L]$ inside the bulk, the so-called full-counting-statistics (FCS). For both observables, we compute, for large $N$, the distribution of the typical as well as atypical large fluctuations. We show that the distribution of the typical fluctuations of the gap are described by the scaling form ${\cal P}_{\rm gap, bulk}(g,N) \sim N H_\alpha(g\,N)$, where $\alpha$ is the interaction coupling and the scaling function $H_\alpha(z)$ is computed explicitly. It has a faster than Gaussian tail for large $z$: $H_\alpha(z) \sim e^{-z^3/(96 \alpha)}$ as $z \to \infty$. Similarly, for the FCS, we show that the distribution of the typical fluctuations of $N_I$ is described by the scaling form ${\cal P}_{\rm FCS}(N_I,N) \sim 2\alpha \, U_\alpha[2 \alpha(N_I - \bar{N}_I)]$, where $\bar{N}_I = L\,N/(2 \alpha)$ is the average value of $N_I$ and the scaling function $U_\alpha(z)$ is obtained explicitly. For both observables, we show that the probability of large fluctuations are described by large deviations forms with respective rate functions that we compute explicitly. Our numerical Monte-Carlo simulations are in good agreement with our analytical predictions.