Extremes of Chi triangular array from the Gaussian $\beta$-Ensemble at high temperature

We study the extreme point process associated to the off-diagonal components in the matrix representation of the Gaussian $\beta$-Ensemble and prove its convergence to Poisson point process as $n\to +\mathrm{\infty }$ when the inverse temperature $\beta$ scales with $n$ and tends to $0$. We consider two main high temperature regimes: ${\displaystyle \beta \ll \frac{1}{n}}$ and ${\displaystyle n\beta =2\gamma \ge 0}$. The normalizing sequences are explicitly given in each cases. As a consequence, we estimate the first order asymptotic of the largest eigenvalue of the Gaussian $\beta$-Ensemble.