Borot–Eynard–Majumdar–Nadal asymptotic conjecture for the generalized Tracy–Widom distribution

Let $F_{\beta}(t)$ denote the generalized Tracy–Widom distribution for $\beta>0$. As $t\to-\infty$, let $\chi$ be the constant term in its left-tail asymptotic expansion. The functions $\zeta(z)$ and $\gamma$ denote the Riemann zeta-function and Euler's constant, respectively.

Borot–Eynard–Majumdar–Nadal conjecture. The left-tail asymptotic expansion is

$$F_{\beta}(t)=\exp\left(-\beta\frac{|t|^3}{24}+\frac{\sqrt{2}}{3}\left(\frac{\beta}{2}-1\right)|t|^{3/2}+\frac{1}{8}\left(\frac{\beta}{2}+\frac{2}{\beta}-3\right)\log|t|+\chi+O\left(\frac{1}{|t|^{3/2}}\right)\right),\qquad t\to-\infty,$$

where

$$\chi=\frac{\beta}{2}\left(\frac{1}{12}-\zeta'(-1)\right)+\frac{\gamma}{6\beta}-\frac{\log 2\pi}{4}-\frac{\log(\beta/2)}{2}+\left(\frac{17}{8}-\frac{25}{24}\left(\frac{\beta}{2}+\frac{2}{\beta}\right)\right)\log 2+\int_{0}^{\infty}\frac{1}{e^{\beta t/2}-1}\left(\frac{t}{e^t-1}-1+\frac{t}{2}-\frac{t^2}{12}\right)\frac{dt}{t^2}.$$

This is a principal heuristic prediction for the asymptotics of generalized Tracy–Widom distributions beyond the classical values $\beta=1,2,4$. The orthogonal-polynomial method available in the classical random-matrix cases does not extend to general $\beta$, so the prediction remains conjectural in this source.