### Ana Flack ^{1} Satya N. Majumdar ^{1} Grégory Schehr ^{2} Satya Majumdar ^{1}

#### Ana Flack, Satya N. Majumdar, Grégory Schehr, Satya Majumdar. Gap probability and full counting statistics in the one-dimensional one-component plasma. *Journal of Statistical Mechanics: Theory and Experiment*, IOP Publishing, 2022, 2022 (5), pp.053211. ⟨10.1088/1742-5468/ac6a59⟩. ⟨hal-03832000⟩

Abstract We consider the 1 d one-component plasma 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 g is described by the scaling form P gap,bulk ( g , N ) ∼ N H α ( g N ) , where α is the interaction coupling and the scaling function H α ( z ) is computed explicitly. It has a faster than Gaussian tail for large z : H α ( z ) ∼ e − z 3 / ( 96 α ) as z → ∞. Similarly, for the FCS, we show that the distribution of the typical fluctuations of N I is described by the scaling form P FCS ( N I , N ) ∼ 2 α U α [ 2 α ( N I − N ¯ I ) ] , where N ¯ I = L N / ( 2 α ) is the average value of N I and the scaling function U α ( z ) is obtained explicitly. For both observables, we show that the probability of large fluctuations is 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.

- 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
- 2. LPTHE – Laboratoire de Physique Théorique et Hautes Energies