Prähofer–Spohn conjecture on TASEP height fluctuation distributions

Let $\rho_-$ and $\rho_+$ be the left and right Bernoulli densities for the TASEP, let $\bar h(y)$ be the hydrodynamic limiting height profile, and let $h_t(j)$ be the height at time $t$ and position $j$. Write $\rho$ for a stationary density, let $y_c$ be the cusp location when $\rho_-<\rho_+$, and let $F_{\mathrm{G}}$, $F_{\mathrm{GUE}}$, $F_{\mathrm{GOE}}$, and $F_0$ denote the distribution functions specified in the source. Prähofer–Spohn conjecture. The centered height fluctuations have the following limiting laws: in the two linear regions, after the stated $t^{1/2}$ normalization, they converge to $F_{\mathrm{G}}$; at the cusp they converge to the product of two Gaussian distribution functions; in the curved region, after the stated $t^{1/3}$ normalization, they converge to $F_{\mathrm{GUE}}$; on either critical line they converge to $F_{\mathrm{GOE}}(x)^2$; and at the merger point $\rho_-=\rho_+=\rho$, $y=1-2\rho$, they converge to $F_0$. In particular, the Gaussian case with $\rho_-=\rho_+$ and the GUE case with $\rho_-=1$, $\rho_+=0$ are proved, while the remaining cases are not established here. This conjectural classification describes the distinct fluctuation universality classes of TASEP and their relation to the geometry of the hydrodynamic height profile.