Leslie Molag: A pluricomplex error-function kernel at the edge of polynomial Bergman kernels

Abstract: The eigenvalues of random normal matrices are known to form a determinantal point process, where the corresponding correlation kernel has the interpretation of a polynomial Bergman kernel depending on a potential. Using polynomial Bergman kernels on \(\mathbb C^d\), constructed with multivariate complex orthogonal polynomials, these determinantal point processes can be extended to higher complex dimension \(d>1\). This setting was initiated by Berman about two decades ago, who derived far-reaching results, in particular in the bulk, but explicitly left the local edge universality as a challenging open problem. We show that the local edge universality observed for \(d=1\), proved in generality by Hedenmalm and Wennman, involving the so-called error-function (or Faddeeva) kernel, extends to \(d>1\). We prove this for two qualitatively different classes, while the full edge universality is still open. Furthermore, we show that, after proper rescaling and rotation, there is a novel universal kernel, which is a multivariate generalization of the error-function kernel. We explicitly identify the subspace of the Bargmann-Fock space where the new kernel is reproducing. Finally, we also present some edge universality results on counting statistics for \(d>1\).