Truncated linear statistics in the one dimensional one-component plasma

Abstract

In this paper, we study the probability distribution of the observable $s=(1/N){\sum }_{i=N-N^{\prime }+1}^N{x}_i$, with $1\le N^{\prime }\le N$ 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 $N^{\prime }=\kappa N$ (with $0<\kappa \le 1$) 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 $⟨s⟩$ 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 ${\mathcal{P}}_{N,\kappa }(s)\simeq \exp \left[-N^3{\varphi }_{\kappa }(s)\right]$, where the rate function ${\varphi }_{\kappa }(s)$ is computed analytically. We show that ${\varphi }_{\kappa }(s)$ 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 $\varphi (\kappa ,s)$ 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.

arXiv comment36 pages, 9 figures

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