Decorrelation transition in the Wigner minor process.

Abstract

We consider the Wigner minor process, i.e. the eigenvalues of an $N\times N$ Wigner matrix $H^{\left(N\right)}$ together with the eigenvalues of all its $n\times n$ minors, $H^{\left(n\right)}$ , $n\le N$ . The top eigenvalues of $H^{\left(N\right)}$ and those of its immediate minor $H^{(N-1)}$ are very strongly correlated, but this correlation becomes weaker for smaller minors $H^{(N-k)}$ as k increases. For the GUE minor process the critical transition regime around $k\sim N^{2/3}$ was analyzed by Forrester and Nagao (Forrester and Nagao, J. Stat. Mech. (08): P08011, 2011) providing an explicit formula for the nontrivial joint correlation function. We prove that this formula is universal, i.e. it holds for the Wigner minor process. Moreover, we give a complete analysis of the sub- and supercritical regimes both for eigenvalues and for the corresponding eigenvector overlaps, thus we prove the decorrelation transition in full generality.