LAUSR

In this paper, we study the probability distribution of the observable s=(1/N)∑i=N−N′+1Nxi , with 1 ⩽ N′ ⩽ N and x 1 < x 2 <⋯< 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 proportional to the center of mass of the N′ = κN (with 0 < κ ⩽ 1), rightmost particles. It interpolates between the position of the rightmost particle (in the limit κ → 0) and the full center of mass (in the limit κ → 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 κ, are instead described by a large deviation form PN,κ(s)≃exp−N3ϕκ(s) , where the rate function ϕ κ (s) is computed analytically. We show that ϕ κ (s) takes different functional forms in five distinct regions in the (κ, s) plane separated by phase boundaries, thus leading to a rich phase diagram in the (κ, s) plane. Across all the phase boundaries the rate function ϕ κ (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.