Uniqueness of the Gaussian Orthogonal Ensemble

Jose Angel Sanchez Gomez, Victor Amaya Carvajal

Published 2019-01-26Version 1

It is not hard to prove that, given a random Wigner matrix $X$ that belongs to the Gaussian Orthogonal Ensemble (GOE), then $X$ is invariant under orthogonal conjugations. Our aim is to prove that if $X$ is a random symmetric matrix that is invariant under orthogonal transformations then such matrix $X$ belongs to the GOE. We will to this using nothing more than the characteristic function.