Linear statistics for Coulomb gases: higher order cumulants

We consider $N$ classical particles interacting via the Coulomb potential in spatial dimension $d$ and in the presence of an external trap, at equilibrium at inverse temperature $\beta$. In the large $N$ limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form ${\cal L}_N = \sum_{i=1}^N f({\bf x}_i)$, where ${\bf x}_i$'s are the positions of the particles and where $f({\bf x}_i)$ is a sufficiently regular function. There exists at present standard results for the first and second moments of ${\cal L}_N$ in the large $N$ limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of ${\cal L}_N$ at large $N$, when the function $f({\bf x})=f(|{\bf x}|)$ and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of $f'(|{\bf x}|)$ and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a $d$-dimensional sphere. In the particular two-dimensional case $d=2$ at the special value $\beta=2$, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of ${\cal L}_N$.