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We study the planar Coulomb gas in the regime where the inverse temperature$\beta_n$ grows at least logarithmically with respect to the number ofparticles $n$ (freezing regime, $\beta_n\gtrsim \log n$). We show that, almostsurely for large $n$, the discrepancy between the number of particles in anymicroscopic region and their expected value (given with adequate precision bythe equilibrium measure) is, up to log factors, of the order of the perimeterof the observation window. The estimates are valid throughout the whole droplet(the region where the particles accumulate), and are particularly interestingnear the boundary, while in the bulk they offer technical improvements overknown results.  Our work builds on recent results on equidistribution at low temperatures andimproves on them by providing refined spectral asymptotics for certain Toeplitzoperators on the range of the erfc-kernel (sometimes called Faddeeva or plasmadispersion kernel).

Improved discrepancy for the planar Coulomb gas at low temperatures