Li: Soft edge limit of the Laguerre beta-ensemble at the lower edge

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.