Law of fractional logarithm and decorrelation transition in the Wigner minor process
A talk in the Seminar Zufallsmatrizen series by
Oleksii Kolupaiev
| Abstract: | I will talk about the two related results mentioned in the title, both of them concern the Wigner minor process, which is a sequence of appropriately scaled $N\times N$ upper left corners of a doubly infinite symmetric array of i.i.d. random variables. We establish the analogue of the Hartman-Wintner law of iterated logarithm for the top eigenvalue of these matrices. This result was initially coined as a law of fractional logarithm (LFL) by E. Paquette and O. Zeitouni, who resolved the special case of GUE matrices. Our work verifies the 10-year-old conjecture by these authors, proving the LFL in full generality for both symmetry classes. Additionally, we show that the correlation between the top eigenvalues in the minor process becomes weaker as the difference between the sizes of the minors increases. We establish the precise description of the resulting decorrelation transition, extending the result of J. Forrester and T. Nagao for the GUE case. The talk is based on the recent joint works with Z. Bao, G. Cipolloni, L. Erd{\H o}s and J. Henheik. Within the CRC this talk is associated to the project(s): C6 |