Tracy–Widom GUE fluctuations for half-space Beta RWRE point-to-point probabilities

Let $\theta>\mu>0$ and $\eta>0$. Let $\mathsf P_{0,2t}(x_\theta t,1)$ be the half-space Beta RWRE point-to-point probability. Define

$$x_\theta=\frac{2\psi_1(\theta)-\psi_1(\theta+\mu)-\psi_1(\theta-\mu)}{\psi_1(\theta+\mu)-\psi_1(\theta-\mu)},\qquad a_\theta=-G'(\theta),\qquad b_\theta=\left(\frac{G”'(\theta)}{2}\right)^{\frac13},$$

where $\psi_1(z)=\partial_z^2\log\Gamma(z)$ and

$$G(z)=\log\left(\frac{\Gamma(z)^2}{\Gamma(z+\mu)\Gamma(z-\mu)}\right)+x(\theta)\log\left(\frac{\Gamma(z-\mu)}{\Gamma(z+\mu)}\right).$$

Tracy–Widom GUE fluctuation conjecture. One has

$$\lim_{t\to\infty}\mathbb P\left(\frac{\log\mathsf P_{0,2t}(x_\theta t,1)-a_\theta t}{b_\theta t^{1/3}}\leqslant y\right)=F_{\rm GUE}(y),$$

where $F_{\rm GUE}$ is the Tracy–Widom GUE distribution function. This is a non-rigorous KPZ-universality prediction for the large-deviation point-to-point probability; the source describes the asymptotic analysis as non-rigorous and does not establish the limit.