13 problems

Let an Airy line ensemble be a non-intersecting line ensemble with the Brownian Gibbs property, and suppose that after subtracting a parabola its law is stationary. A height shift…

Let $A(\tau)$ be the top curve of the Airy process, and let $\xi_1,\ldots,\xi_m$ and $\tau_1,\ldots,\tau_m$ specify its $m$-dimensional distribution functions … For $m=1$, the corr…

Uniqueness conjecture. For each $T>0$, $H(t)$ has a unique point of maximum in $[-T,T]$ almost surely.

Let $D$ be the space of admissible directed-landscape environments, let $I$ be the associated rate function, and let $Q$ be the Dirichlet energy of a function…

Let $D^{\mathrm{Br}}$ denote the Brownian web distance, let $\mathfrak A(y)=\mathcal L(0,0;1,y)$ be the parabolic Airy process associated with the directed landscape $\mathcal L$,…

Let $\mathcal{L} = (\mathcal{L}_1, \mathcal{L}_2, \ldots)$ be a line ensemble such that $\mathcal{L}(x)+x^2$ is stationary under deterministic horizontal shifts, $\mathcal{L}$ has…

Fix $\theta,r>0$ and let $\tilde{f}_N^{LG}$ be the random continuous function from Definition 2. Then, as $N\to\infty$, the random functions $\tilde{f}_N^{LG}$ converge weakly in…

Let $h(t,x)$ denote the solution of the KPZ equation with narrow wedge initial data, and consider its fluctuations under horizontal scaling by $t^{2/3}$, vertical scaling by…

Higher-eigenvalue Airy-process conjecture. For any fixed $k$, as $n\to\infty$, the tuple $(\alpha_{n,1},\ldots,\alpha_{n,k})$ converges in distribution to the top $k$ points of the…

Let $\mathcal{F}(T,X)$ denote the narrow-wedge KPZ equation Hopf–Cole solution, and let $T>0$ and $X\in\mathbb{R}$. Corwin–Quastel conjecture. As $T\to\infty$, … converges to the A…

Johansson's conjecture. Almost surely, $H(t)$ has a unique maximum point in $[-T,T]$.

In the continuum random polymer, write the stochastic heat-equation solution in the form … Here $A_t$ is the resulting crossover process and $\mathcal{A}_2$ denotes the Airy$_2$ pr…

Let $mathcal{F}(T,X)$ denote the continuum directed random polymer free energy at time $T$ and spatial position $X$, and let ${mathscr{A}}2(X)$ be the ${rm Airy}2$ process. **Airy…