Advanced Search

The Tracy–Widom $\beta$ distribution is the large dimensional limit of the top eigenvalue of $\beta$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a\to \infty$ the tail of the Tracy–Widom distribution satisfies $$P({\mathrm{𝑇𝑊}}_{\beta }gt;a)=a^{-(3/4)\beta +\mathrm{o}\left(1\right)}exp\left(-\frac{2}{3}\beta a^{3/2}\right).$$