Correlation functions between eigenvalues and singular values
A talk in the Seminar Zufallsmatrizen series by
Matthias Allard
| Abstract: | The talk is about the probabilistic relations between eigenvalues and singular values of bi-unitarily invariant ensembles. We first extend the notion of k-point correlation function to j,k-point correlation functions when studying the interactions between j singular values and k singular values and, then, give an exact formula for the 1,1-point correlation function. This formula simplifies drastically when assuming the singular values are drawn from a polynomial ensemble. We will give some idea of the proof for the main result. Finally, we will show some numerical simulations to illustrate what the 1,1-point correlation function looks like for the classical cases of Laguerre and Jacobi ensembles and what it reveals about the interactions between singular values and eigenvalues. Within the CRC this talk is associated to the project(s): C6 |