Edge universality of random normal matrices generalizing to higher dimensions
A talk in the Seminar Zufallsmatrizen series by
Leslie Diëgo Molag
| Abstract: | As recently proved in generality by Hedenmalm and Wennman, it is a universal behavior of
complex random normal matrix models that one finds a complementary error function
behavior at the boundary (also called edge) of the droplet as the matrix size increases.
Such behavior is seen both in the density, and in the off-diagonal case, where the
Faddeeva plasma kernel emerges. These results are neatly expressed with the help of the
outward unit normal vector on the edge. We prove that such universal behaviors transcend
this class of random normal matrices, being also valid in higher dimensional determinantal
point processes, defined on $\mathbb C^d$. The models under consideration concern higher
dimensional generalizations of the determinantal point processes describing the
eigenvalues of the complex Ginibre ensemble and the complex elliptic Ginibre ensemble.
These models describe a system of particles with mutual repulsion, that are confined to the
origin by an external field $V(z)=|z|^2 - \tau\Re(z_1^2+...+z_d^2)$, where $0\leq \tau < 1$. Their
average density of particles converges to a uniform law on a 2d-dimensional ellipsoidal
region. It is on the boundary of this region that we find a complementary error function
behavior and the Faddeeva plasma kernel. To the best of my knowledge, this is the first
instance of the Faddeeva plasma kernel emerging in a higher dimensional model. Based
on arXiv:2208.12676 Please contact Gernot Akemann (akemann@physik.uni-bielefeld.de) for details regarding access. Within the CRC this talk is associated to the project(s): C6 |