Linear statistics at the microscopic scale for the 2D Coulomb gas

We consider the classical Coulomb gas in two dimensions at the inverse temperature β=2, confined within a droplet of radius R by a rotationally invariant potential $U(r)$. For U(r)~ r² this describes the eigenvalues of the complex Ginibre ensemble of random matrices. We study linear statistics of the form LN = ∑ᵢ₌₁N f(|ᵢ|), where ᵢ's are the positions of the N particles, in the large N limit with $R=O(1)$. It is known that for smooth functions $f(r)$ the variance Var \, LN= O(1), while for the indicator function f(r)= I0<r< r with 0< r ≤ R, relevant for the disk counting statistics, all cumulants of LN of order q ≥ 2 behave as ~ √N. In addition, for smooth functions, it was shown recently that the cumulants of LN of order q ≥ 3 scale as O(N2-q). Surprisingly it was found that they depend only on f'(| x|) and its derivatives evaluated exactly at the boundary of the droplet. To understand this property, and interpolate between the two behaviors (smooth versus step-like), we study the microscopic linear statistics given by f(r) → fN(r) = φ((r- r) √N/ξ), which probes the fluctuations at the scale of the inter-particle distance. We compute the cumulants of LN at large N for a fixed shape function φ(u) at arbitrary ξ. For large ξ they match the predictions for smooth functions, showing that the leading contribution in that case comes from a boundary layer of size 1/√N near the boundary of the droplet. Finally we show that the distribution of~LN takes two distinct large deviation forms, in the regime LN = O(√N) and LN =O(N) respectively. The transition between these two regimes is accompanied by the formation of a macroscopic hole in the distribution of charges.