Truncated linear statistics in the one dimensional one-component plasma

In this paper, we study the probability distribution of the observable $s = (1/N)\sum_{i=N-N'+1}^N x_i$, with $1 \leq N' \leq N$ and $x_1<x_2<\cdots< x_N$ representing the ordered positions of $N$ particles in a $1d$ one-component plasma, i.e., $N$ harmonically confined charges on a line, with pairwise repulsive $1d$ Coulomb interaction $|x_i-x_j|$. This observable represents an example of a truncated linear statistics -- here the center of mass of the $N' = \kappa \, N$ (with $0 < \kappa \leq 1$) rightmost particles. It interpolates between the position of the rightmost particle (in the limit $\kappa \to 0$) and the full center of mass (in the limit $\kappa \to 1$). We show that, for large $N$, $s$ fluctuates around its mean $\langle s \rangle$ and the typical fluctuations are Gaussian, of width $O(N^{-3/2})$. The atypical large fluctuations of $s$, for fixed $\kappa$, are instead described by a large deviation form ${\cal P}_{N, \kappa}(s)\simeq \exp{\left[-N^3 \phi_\kappa(s)\right]}$, where the rate function $\phi_\kappa(s)$ is computed analytically. We show that $\phi_{\kappa}(s)$ takes different functional forms in five distinct regions in the $(\kappa,s)$ plane separated by phase boundaries, thus leading to a rich phase diagram in the $(\kappa,s)$ plane. Across all the phase boundaries the rate function $\phi(\kappa,s)$ undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.