Speaker
Li, University of Wisconsin-Madison
Abstract
We show that the appropriately scaled lower edge of the Laguerre beta-ensemble with fixed \($\beta$\), \($a=a_n\to \infty$\), and \($a/n\to 0$\) converges to the \($Airy_{\beta}$\) point process. The methods of Rider, Ramírez, and Virág can be used to prove this statement when \($\liminf a_n/n>0$\), but they do not apply in our regime of parameters. When \($a_n\gg (\log\log n)^3$\), we prove a stronger, operator level version of the convergence. When \($a_n\le (\log n)^{1/2}$\), we use a different argument that relies on coupling and the hard-to-soft process level transition between the hard and soft edge limit processes.