Harmonic Analysis and Signal Processing Seminar
Universality and hyperuniformity of Weyl-Heisenberg ensembles
Luis Daniel Abreu
Tuesday, May 2, 2017, 2pm, WWH 201
The Weyl-Heisenberg ensemble is a functional dependent class of
determinantal point process (DPPs) associated with the Schrödinger
representation of the Heisenberg group. In contrast with other DPPs, for
most choices of the function g, one has no access to explicit formulas for
the correlation kernel. To overcome this obstruction, a new methodology
has been developed, based on phase space and operator theoretical
principles. We will show that infinite Weyl-Heisenberg ensembles are
hyperuniform, that the asymptotic limit of its finite-dimensional
counterparts satisfy a new universality property (which can be seen as a
geometric variation of the circular law) and obtain sharp estimates for
the corresponding rate of convergence.
By properly selecting the function g, we recover known results for the
Ginibre ensemble and obtain natural extensions for its higher Landau
levels counterparts known as polyanalytic Ginibre ensembles.
The talk is based on joint work with K. Gröchenig, J. M. Pereira, J. L.
Romero and S. Torquato.