Large deviations principle for the largest eigenvalue of the Gaussian betaensemble at high temperature
Abstract
We consider the Gaussian betaensemble when $\beta$ scales with $n$ the number of particles such that $\displaystyle{{n}^{1}\ll \beta\ll 1}$. Under a certain regime for $\beta$, we show that the largest particle satisfies a large deviations principle in $\mathbb{R}$ with speed $n\beta$ and explicit rate function. As a consequence, the largest particle converges in probability to $2$, the rightmost point of the semicircle law.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.07651
 Bibcode:
 2018arXiv180607651P
 Keywords:

 Mathematics  Probability;
 60B20;
 60F10
 EPrint:
 16 pages