Characteristic Polynomials of the Hermitian Wigner and Sample Covariance Matrices

Joint Princeton Mathematical Physics Seminar
Topic:Characteristic Polynomials of the Hermitian Wigner and Sample Covariance Matrices
Speaker:Tatyana Shcherbina
Affiliation:Institute for Low Temperature Physics, Kharkov
Date:Tuesday, November 1
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/joint/shcherbina

We consider asymptotics of the correlation functions of characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$ and the hermitian sample covariance matrices $X_n=n^{-1}A_{m,n}^*A_{m,n}$. We use the integration over the Grassmann variables to obtain a convenient integral representation. Then we show that the asymptotics of the correlation functions of any even order coincide with that for the GUE up to a factor, depending only on the fourth moment of the common probability law of the matrix entries, i.e. that the higher moments do not contribute to the above asymptotics.