15 problems

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

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…

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…

The parabolic Airy process is … where $\mathcal{A}_2$ is the Airy$_2$ process. Universality conjecture. The parabolic Airy process is conjectured to be a universal limit of models…

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

Johansson's conjecture. The function $\mathfrak{h}_1(\cdot;\delta_0)$ has a unique argmax almost surely.

The Airy$_2$ process is associated with delta initial conditions for the KPZ equation, and other initial profiles should be obtainable by integrating this delta initial condition.…

Let $\,^{(\tau)}\!\overset{\mathrm{inv},1}{H}$ denote the time-dependent matrix process with orthogonal symmetry, and let $_M\!\!\overset{\mathrm{inv},1}\Lambda(s,0)$ be its edge-s…

The Airy$_2$ time-dependent point process is the determinantal process arising as the edge-scaling limit of suitable random matrix and growth-model ensembles. It is conjectured to…

Airy$_2$ fluctuation conjecture. There exists $\sigma_1>0$ such that

Space-time scaling-limit conjecture. This space-time process is the scaling limit of last-passage percolation models and of solutions to the Kardar–Parisi–Zhang equation. The conje…

Airy$_2$ sheet conjecture. The process $\mathcal A_n(u,v)$ converges to a two-parameter process $(\mathcal A(u,v),(u,v)\in\mathbb R^2)$, called the Airy$_2$ sheet. The sequence is…

Let $\mathcal{T}$ be the location of the maximizer of $\mathcal{A}_2(t)-t^2$, whose almost-sure uniqueness is established in the surrounding discussion. The notation $e^{-ct^3}$ de…

Let $z(0,x)=e^{B(x)}\mathbf{1}_{x\geq0}$, where $B(x)$ is a standard Brownian motion, and let $\mathcal{A}_{2\to\mathrm{BM}}$ denote the Airy crossover process appearing in the sca…

The six basic initial-data classes in KPZ growth are associated with the Airy processes $\operatorname{Airy}_2$, $\operatorname{Airy}_1$, $\operatorname{Airy}_{\rm stat}$,…