Hard edge tail asymptotics

Abstract

Let $\Lambda$ be the limiting smallest eigenvalue in the general $(\beta,a)$-Laguerre ensemble of random matrix theory. That is, $\Lambda$ is the $n\to\infty$ distributional limit of the (scaled) minimal point drawn from the density proportional to $\Pi_1\leq i\leq j\leq n$ $$\left|\lambda_i-\lambda_j\right|^\beta\prod_{i=1}^n\lambda_i^{\frac{\beta}{2}(a+1)-1}e^{-\frac{\beta}{2}\lambda_i}$$ on $(\mathbb{R}_+^n$. Here $\beta>0$, $a> -1$; for $\beta=1,2,4$ and integer $a$, this object governs the singular values of certain rank $n$ Gaussian matrices. We prove that $$ \mathbb{P}(\Lambda>\lambda)=e^{-\frac{\beta}{2}\lambda+2\gamma\sqrt{\lambda}}\lambda^{-\frac{\gamma(\gamma+1-\beta/2)}{2\beta}} e(\beta,a)(1+o(1))$$ as $\lambda\to\infty$ in which $$\gamma = \frac{\beta}{2} (a+1)-1$$ and $e(\beta, a) > 0$ is a constant (which we do not determine). This estimate complements/extends various results previously available for special values of $\beta$ and $a$.