Cumulants and Large Deviations for the Linear Statistics of the One-Dimensional Trapped Riesz Gas

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 (\(k \rightarrow 0\)) 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 \(L/\ell _0 \rightarrow 1\), which we relate to the large deviations of the emptiness probability of the complementary interval on the real line.