Birth of a gap: Critical phenomena in 2D Coulomb gas

We investigate a family of radially symmetric Coulomb gas systems at inverse temperature $\beta = 2$. The family is characterised by the property that the density of the equilibrium measure vanishes on a ring at radius $r_*$, which lies strictly inside the droplet. The large-$n$ expansion of the logarithm of the partition function is obtained up to the constant term and features a novel term of order $n^{1/4}$. We perform a double scaling limit of the correlation kernel at the $n^{1/4}$ scale and obtain a new limiting kernel in the bulk, which differs from the well-known Ginibre kernel.