Edge universality of deformed non-Hermitian random matrices
A talk in the Seminar Zufallsmatrizen series by
Andrew Campbell
| Abstract: | For an $N\times N$ Hermitian matrix $A$ the eigenvalues of the top-left $N-
1\times N-1$ submatrix (or truncation) of $A$ interlace with the original
eigenvalues of $A$. We could then continue to remove rows and columns to get
further interlacing sequences of eigenvalues, and we can think of this process
as some kind of dynamics on the spectral measures. Similarly differentiating
real rooted polynomials will produce interlacing among the roots, and we can
think of this as some other dynamics on the root measures. Various recent
results have shown, both heuristically and rigorously, that for random
matrices these two processes produce the same dynamics on the measures.
However, if we consider the analogous processes from non-Hermitian matrices
or complex rooted polynomials there is no obvious geometric reason for the
processes to coincide and the picture is much less clear. After looking at a brief
history of the real case, we will discuss how one can connect these processes
for single ring matrices and random polynomials with independent
coefficients. This talk will be based on joint work with Sean O'Rourke and
David Renfrew. Please contact Lucas Hackl (Lucas.Hackl@unimelb.edu.au) for details regarding access Within the CRC this talk is associated to the project(s): C6 |