Improved discrepancy for the planar Coulomb gas at low temperatures

We study the planar Coulomb gas in the regime where the inverse temperature $\beta_n$ grows at least logarithmically with respect to the number of particles $n$ (freezing regime, $\beta_n\gtrsim \log n$). We show that, almost surely for large $n$, the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results. Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel).