Cumulants and large deviations for the linear statistics of the one-dimensional trapped Riesz gas

Abstract

We consider the classical trapped Riesz gas, i.e., $N$ particles at positions $x_i$ in one dimension with a repulsive power law interacting potential $\propto 1/|x_i-x_j{|}^k$, with $k>-2$, in an external confining potential of the form $V(x)\sim |x{|}^n$. We focus on the equilibrium Gibbs state of the gas, for which the density has a finite support $[-{\ell }_0/2,{\ell }_0/2]$. We study the fluctuations of the linear statistics ${\mathcal{L}}_N={\sum }_{i=1}^{N}f(x_i)$ in the large $N$ limit for smooth functions $f(x)$. We obtain analytic formulae for the cumulants of ${\mathcal{L}}_N$ for general $k>-2$. For long range interactions, i.e. $k<1$, which include the log-gas ($k\to 0$) and the Coulomb gas ($k=-1$) these are obtained for monomials $f(x)=|x{|}^m$. For short range interactions, i.e. $k>1$, which include the Calogero-Moser model, i.e. $k=2$, we compute the third cumulant of ${\mathcal{L}}_N$ for general $f(x)$ and arbitrary cumulants for monomials $f(x)=|x{|}^m$. We also obtain the large deviation form of the probability distribution of ${\mathcal{L}}_N$, which exhibits an "evaporation transition" where the fluctuation of ${\mathcal{L}}_N$ is dominated by the one of the largest $x_i$. In addition, in the short range case, we extend our results to a (non-smooth) indicator function $f(x)$, obtaining thereby the higher order cumulants for the full counting statistics of the number of particles in an interval $[-L/2,L/2]$. We show in particular that they exhibit an interesting scaling form as $L/2$ approaches the edge of the gas $L/{\ell }_0\to 1$, which we relate to the large deviations of the emptiness probability of the complementary interval on the real line.

36 pages, 1 figure, 2 tables