Workshop: Linear Statistics for Coulomb and Riesz gases: higher order cumulants

Speaker
Gregory Schehr, LPTHE, Sorbonne University

Abstract
I will first consider $N$ classical particles interacting via the Coulomb potential in spatial dimension $d$ and in the presence of an external trap, at equilibrium. For large $N$, the particles are confined within a droplet of finite size. The main focus here is on smooth linear statistics, i.e. the fluctuations of sums of the form ${\cal L}_N = \sum_{i=1}^N f(\bx_i)$, where $\bx_i$’s are the positions of the particles and where $f(\bx_i)$ 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 $f(\bx)=f(|{\bf x}|)$ 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 {\it evaluated exactly at the boundary of the droplet}, which in this case is a $d$-dimensional sphere. I will then extend these results to the linear statistics of one-dimensional trapped Riesz gases, 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 $V(x) \sim |x|^n$.