4 problems

Let $\lambda_{\max}^{(N)}(t)$ be the largest eigenvalue of an $N\times N$ matrix undergoing Dyson's Brownian motion with parameter $\beta=1$. Let ${\cal A}_{\rm 1}$ denote the Airy…

Let $h(x,t)$ be the height function of the polynuclear growth (PNG) model on a one-dimensional substrate with flat initial conditions. Let ${\cal A}_{\rm 1}$ denote the Airy one pr…

Universal Airy-one scaling conjecture. There exist universal constants $c_1$ and $c_2$ such that

Johansson's conjecture. Almost surely, $x\mapsto\mathcal{A}(x)-x^2$ attains its maximum at a unique point $Y\in\mathbb{R}$.