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

Abstract

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 ${\mathcal{P}}_{\mathrm{gap},\mathrm{bulk}}(g,N)\sim N{H}_{\alpha }(gN)$, 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 ${\mathcal{P}}_{\mathrm{FCS}}(N_I,N)\sim 2\alpha U_{\alpha }[2\alpha (N_I-{\bar{N}}_I)]$, where ${\bar{N}}_I=LN/(2\alpha )$ 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.

32 pages, 8 figures

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