Uniqueness of the Gaussian Orthogonal Ensemble

Jose Angel Sanchez Gomez; Victor Amaya Carvajal

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arXiv · 1901.09257

Uniqueness of the Gaussian Orthogonal Ensemble

Jose Angel Sanchez Gomez·Victor Amaya Carvajal

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Abstract

A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The goal of this work is to prove the converse, that is, if $X$ is a symmetric random matrix such that it is invariant under orthogonal conjugations, then such matrix $X$ belongs to the GOE. We will prove this using some elementary properties of the characteristic function of random variables.

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Jose Angel Sanchez Gomez, Victor Amaya Carvajal. 2019-01-26. Uniqueness of the Gaussian Orthogonal Ensemble. https://arxiv.org/abs/1901.09257

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