Poisson statistics at the edge of Gaussian $\beta$-ensemble at high temperature

Cambyse Pakzad (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We study the asymptotic edge statistics of the Gaussian $\beta$-ensemble, a collection of $n$ particles, as the inverse temperature $\beta$ tends to zero as $n$ tends to infinity. In a certain decay regime of $\beta$, the associated extreme point process is proved to converge in distribution to a Poisson point process as $n\to +\infty$. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.

Domaines: Probabilités [math.PR]

Cambyse Pakzad. Poisson statistics at the edge of Gaussian $\beta$-ensemble at high temperature. 2018. ⟨hal-01777520⟩