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

Abstract

We consider the classical Coulomb gas in two dimensions at the inverse temperature $\beta =2$, 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 ${\mathcal{L}}_N={\sum }_{i=1}^{N}f(|{\mathbf{x}}_i|)$, where ${\mathbf{x}}_i$'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 $\mathrm{Var}{\mathcal{L}}_N=O(1)$, while for an indicator function relevant for the disk counting statistics, all cumulants of ${\mathcal{L}}_N$ of order $q\ge 2$ behave as $\sim \sqrt{N}$. In addition, for smooth functions, it was shown that the cumulants of ${\mathcal{L}}_N$ of order $q\ge 3$ scale as $\sim N^{2-q}$. Surprisingly it was found that they depend only on $f^{\prime }(|x|)$ 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 $f(r)\to f_N(r)=\varphi ((r-\hat{r})\sqrt{N}/\xi )$, which probes the fluctuations at the scale of the inter-particle distance. We compute the cumulants of ${\mathcal{L}}_N$ at large $N$ for a fixed $\varphi (u)$ at arbitrary $\xi$. For large $\xi$ 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 ${\mathcal{L}}_N$ take two distinct large deviation forms, in the regime ${\mathcal{L}}_N\sim \sqrt{N}$ and ${\mathcal{L}}_N\sim N$ respectively. We also discuss applications of our results to fermions in a rotating harmonic trap and to the Ginibre symplectic ensemble.

40 pages, 7 figures

References in corpus (27)

• Single-atom imaging of fermions in a quantum-gas microscope
• A Quantum Gas Microscope for Fermionic Atoms
• Site-resolved Imaging of Fermionic Lithium-6 in an Optical Lattice
• Fluctuations of eigenvalues of random normal matrices
• On the structure of Normal Matrix Model
• The Index Distribution of Gaussian Random Matrices
• Random normal matrices and Ward identities
• How many eigenvalues of a Gaussian random matrix are positive?
• Entanglement Entropy and Full Counting Statistics for $2d$-Rotating Trapped Fermions
• The noise in the circular law and the Gaussian free field
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem
• Extremes of $2d$ Coulomb gas: universal intermediate deviation regime
• Entanglement entropy and Berezin-Toeplitz operators
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• Index Distribution of the Ginibre Ensemble
• Quantum Gas Microscopy of Fermions in the Continuum
• Counting statistics for non-interacting fermions in a rotating trap
• Multilayered density profile for noninteracting fermions in a rotating two-dimensional trap
• Universality of the number variance in rotational invariant two-dimensional Coulomb gases
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• Anisotropic Quantum Hall Droplets
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle

Cited by in corpus (1)

• Universality for fluctuations of counting statistics of random normal matrices