# Uniqueness of the Gaussian Orthogonal Ensemble

## TL;DR

This paper proves that symmetric random matrices invariant under orthogonal conjugation must belong to the Gaussian Orthogonal Ensemble, establishing a converse to a known invariance property in random matrix theory.

## Contribution

The work demonstrates that invariance under orthogonal conjugation characterizes matrices in the GOE, providing a new characterization of the ensemble.

## Key findings

- Symmetric matrices invariant under orthogonal conjugation are in the GOE.
- The proof uses properties of the characteristic function of random variables.
- The result completes the understanding of invariance properties in random matrix theory.

## 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.