The planar low temperature Coulomb gas: separation and equidistribution

We consider planar Coulomb systems consisting of a large number $n$ of repelling point charges in the low temperature regime, where the inverse temperature $\beta$ grows at least logarithmically in $n$ as $n \longrightarrow \infty$, i.e., $\beta\gtrsim \log n$. Under suitable conditions on an external potential we prove results to the effect that the gas is with high probability uniformly separated and equidistributed with respect to the corresponding equilibrium measure (in the given external field). Our results generalize earlier results about Fekete configurations, i.e., the case $\beta=\infty$. There are also several auxiliary results which could be of independent interest. For example, our method of proof of equidistribution (a variant of "Landau's method") works for general families of configurations which are uniformly separated and which satisfy certain sampling and interpolation inequalities.