Chen–Manning hard-edge gap probability asymptotic conjecture

Let $E_\beta^{\rm hard}(0;(0,s);a)$ denote the hard-edge probability that $(0,s)$ contains no eigenvalues, with hard-edge parameter $a$. Chen–Manning's conjecture.

$$E_\beta^{\rm hard}(0;(0,s);a) \mathop{\sim}\limits_{s \to \infty} \exp\left(-\frac{\beta s}{8}+a\sqrt{s}-\frac{a^2}{2\beta}\log s+\left(1-\frac{\beta}{2}\right)\frac{a}{2\beta}\log s\right).$$

This prediction comes from the log-gas calculation at the hard edge and describes the large-$s$ asymptotic form of the gap probability.