2 problems

Airy$_2$ conjecture. The fluctuations of $\partial\Phi$ for the typical interface converge to the Airy$_2$ process as $N\to\infty$.

Facet conjecture. With probability tending to $1$ as $\beta\to\infty$, $h(\Gamma)$ is finite and $\Phi(\Gamma)$ is indeed the facet of $\Gamma$, in the sense that the next layer is…