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

TL;DR

The paper analyzes the 2D Coulomb gas at β=2 under a radial confining potential, focusing on linear statistics evaluated at microscopic scales near a reference radius. Employing determinantal structure, it derives exact cumulant generating functions and shows that microscopic fluctuations are governed by a boundary-layer near the droplet edge, with cumulants scaling as √N and expressed via Gaussian averages of a test function φ. It reveals a precise matching between microscopic and macroscopic statistics, and identifies two distinct large-deviation regimes for the full distribution of L_N, including a cubic tail tied to edge-hole formations, with explicit rate functions and a Coulomb-gas constrained-density analysis. The results have implications for Ginibre ensembles, rotating-fermion systems, and related random-matrix models, and they pave the way for extensions to other symmetry classes and multi-component supports.

Abstract

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 $U(r)\sim r^2$ this describes the eigenvalues of the complex Ginibre ensemble of random matrices. We study linear statistics of the form ${\cal L}_N = \sum_{i=1}^N f(|{\bf x}_i|)$, where ${\bf x}_i$'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 ${\rm Var} \,{\cal L}_N= O(1)$, while for an indicator function relevant for the disk counting statistics, all cumulants of ${\cal L}_N$ of order $q \geq 2$ behave as $\sim \sqrt{N}$. In addition, for smooth functions, it was shown that the cumulants of ${\cal L}_N$ of order $q \geq 3$ scale as $\sim N^{2-q}$. Surprisingly it was found that they depend only on $f'(|\bf x|)$ 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 $f(r) \to f_N(r) = φ((r-\hat r) \sqrt{N}/ξ)$, which probes the fluctuations at the scale of the inter-particle distance. We compute the cumulants of ${\cal L}_N$ at large $N$ for a fixed $φ(u)$ at arbitrary $ξ$. For large $ξ$ they match the predictions for smooth functions which shows that the leading contribution in that case comes from a boundary layer of size $1/\sqrt{N}$ near the boundary of the droplet. Finally we show that the full probability distribution of ${\cal L}_N$ take two distinct large deviation forms, in the regime ${\cal L}_N \sim \sqrt{N}$ and ${\cal L}_N \sim N$ respectively. We also discuss applications of our results to fermions in a rotating harmonic trap and to the Ginibre symplectic ensemble.

Figures (7)

• Figure 1: Illustration of the microscopic linear statistics in Eq. (\ref{['def_micro']}) studied in this paper. Here, the blue points correspond to the eigenvalues of an $N \times N$ Ginibre matrices, with $N=500$, described by a Coulomb gas as in Eq. (\ref{['PDF_intro']}) with $U(r)=r^2/2$. In this case the equilibrium density has support on the disk of radius $R$, centered at the origin. The shaded area around the circle of radius $\hat{r}$ and width $\sim \xi/\sqrt{N}$ (depicted in gray scale) indicates the microscopic region (of size comparable to the interparticle spacing) where the linear statistics (\ref{['def_micro']}) takes nonzero values.
• Figure 2: Plot of the conditioned density $\rho_\Lambda({\bf r}) \equiv \rho_\Lambda(r=|{\bf r}|)$ given in Eqs. (\ref{['rho_ansatz']}) and (\ref{['def_I']}). The vertical blue line indicates the delta function at $r=\hat{r}$.
• Figure 3: Sketch of the constrained density profile $\rho_\mu(u)$, illustrating the two main regimes (i) and (ii) described respectively by Eq. (\ref{['first_regime']}) and (\ref{['secondregime']}).
• Figure 4: Illustration of the case where $1 - \hat{\mu} \phi"(v) > 0$ for all $v \in \mathbb{R}$. Left: Plot of $F(v;\hat{\lambda})$ as a function of $v$, which for all $\hat{\lambda}$ exhibits a single minimum at $v^*(\hat{\lambda})$. Right: Plot of $v^*(\hat{\lambda})$ as a function of $\hat{\lambda}$.
• Figure 5: Plot of $F(v;\hat{\lambda})$ vs $v$ in the case where $1 -|\hat{\mu}| \phi"(v) < 0$ for $v \in [v_1, v_2]$ for different values of $\hat{\lambda}$, as explained in the text.