Near-Boundary Asymptotics for Correlation Kernels

We discuss asymptotic formulas for the correlation kernel corresponding to a (more or less) general potential Q in the plane. If K _n is such a kernel, there is a known asymptotic formula K _n (z,z)=nΔQ(z)+B ₁(z)+n ^?1 B ₂(z)+? valid for z in the interior of a certain compact set known as the “droplet” corresponding to Q (on which ΔQ>0). On the other hand, if z is in the exterior of the droplet, K _n (z,z) converges to zero exponentially in n. Results of this type are useful in random matrix theory and conformal field theory; they have recently been used to prove the Gaussian field convergence in the interior of the droplet for fluctuations of eigenvalues of random normal matrices. To be able to extend such results beyond the interior, it becomes necessary to have a certain uniformity of estimates as z approaches the boundary either from the interior or from the exterior. Such estimates have to our knowledge hitherto not been known on a rigorous level. This note intends to fill this gap. We will consider applications in later publications. Our treatment of the (interior) asymptotics relies in an essential way on previous work due to Berman, Berndtsson, and Sjöstrand (Ark. Mat. 46, 2008), and Berman (Indiana Univ. Math. J. 58, 2009). We hope that this note can to some extent be regarded as a contribution to that work.