Gregory Schehr

Edge Effects and Microscopic Scales in Linear Statistics of Coulomb Gases

I will discuss recent progress on the fluctuations of linear statistics in Coulomb gases, highlighting how boundary effects dominate non-Gaussian behaviour. In a general $d$-dimensional rotationally invariant setting, all higher-order cumulants of smooth linear statistics are shown to depend only on the test function's behaviour at the edge of the equilibrium droplet. In the case $d = 2$, $\beta = 2$, these results follow from determinantal techniques. I will then turn to the microscopic regime in two dimensions, where the test function varies on the inter-particle scale. This leads to a crossover between fluctuation regimes and shows how a boundary layer of width $O(N^{-1/2})$ controls the leading cumulants.