Håkan Hedenmalm: Berezin quantization and normal random matrices

We define the Berezin transform in terms of the weighted Bergman kernel for the weight $e^{mQ}$, where $Q$ is a fixed potential and $m$ is a large positive real parameter. Using asymptotical analysis of the weighted Bergman kernel we find that the Berezin transform of a function is essentially the heat extension of the function, for a time which is proportional to $1/m$. We apply the asymptotic analysis to the context of Random Normal Matrices.