Baik–Rains conjecture for the stationary TASEP two-point distribution

Let $\rho ameq{}(0,1)$ be the stationary Bernoulli density, let $\mathbb{P}_\rho$ be the corresponding TASEP path measure, and let $h_t(j)$ be the height function. For $w\in\mathbb{R}$, define the observation point $[(1-2\rho)t+4(\rho(1-\rho))^{1/3}t^{2/3}w]$. Baik–Rains conjecture. For fixed $\rho$ and $w$, the centered and scaled height satisfies

$$\lim_{t\to\infty}\mathbb{P}_\rho\Big(-h_t([(1-2\rho)t+4(\rho(1-\rho))^{1/3}t^{2/3}w])+2\rho(1-\rho)t+(1-2\rho)\big((1-2\rho)t+4(\rho(1-\rho))^{1/3}t^{2/3}w\big)\\ \leq 2(\rho(1-\rho))^{2/3}t^{1/3}x\Big)=F_w(x).$$

The distribution $F_w$ is identified in the cited work with the Baik–Rains distribution $H(4w^2+x;w,-w)$. This describes the stationary KPZ two-point fluctuation profile on the $t^{2/3}$ spatial scale; the source presents it as a conjectural formula, while $F_0$ is also the distribution appearing at the critical point of the preceding classification.