23 problems

Let $L$ be the directed landscape, let $(t)=_{0,0;0,1}(t)$ be the directed-landscape geodesic location, and let $L(t)=L(0,0;(t),t)$ be its segment length. Let…

Directed landscape conjecture. For even larger scales

The directed landscape $\mathcal L$ is a four-parameter scaling-limit object constructed from Brownian last passage percolation. Universality conjecture. It is conjectured to be th…

Directed landscape universality conjecture. The directed landscape is conjectured to be the universal scaling limit for various models in the KPZ universality class.

Let $\Gamma_n$ be the geodesic from $\mathbf{0}$ to $\mathbf{n}$ in exponential last passage percolation, and fix an intermediate time $\gamma\in(0,1)$ with $\gamma\ne1/2$. Under t…

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…

The directed-landscape geodesic upper-tail rate conjecture. For $t\in (0,1/2]$, as $a\to\infty$,

Universality conjecture. The directed landscape is conjectured to be the full scaling limit of all KPZ models, including models of planar first and last passage percolation with on…

A two-dimensional noise is a probability space $(\Omega,\mathcal F,\mathbb P)$ equipped with sub-$\sigma$-algebras associated with open rectangles in $\mathbb R^2$ and measure-pres…

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 $\mathcal L$,…

The Brownian web distance is a directed metric on the tree formed by coalescing simple random walks, and its scale-invariant limit is the Brownian web distance. The full directed l…

Fix $\gamma \in (0,2)$ and consider $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Let $\operatorname{Star}_k$ denote the set of points joined to the origin by at least $k$ geo…

The directed landscape $\mathcal{L}$ is the full scaling limit of a two-dimensional random metric arising from Brownian last passage percolation and other exactly solvable models i…

The KPZ equation describes a random height function, and its large-time behavior is studied through the KPZ fixed point and the KPZ sheet. KPZ fixed-point conjecture. The KPZ fixed…

Let $\mathcal L$ be the directed landscape, let $\pi^*(\tau T)$ be the geodesic location at time $\tau T$, and condition on $\mathcal L(0,0;0,T)=L$. For $\tau\in(0,1)$, consider th…

Geodesic--argmax rigidity conjecture. For every $\tau\in(0,1)$ and $T>0$, conditionally given $\mathcal L(0,0;0,T)=L$, as $L\to\infty$,

Let $\Xi$ denote the exceptional set of directions appearing in the classification of semi-infinite geodesics in the directed landscape. A Busemann geodesic is a semi-infinite geod…

Directed landscape KPZ-universality conjecture. $\mathcal{L}$ is the scaling limit of all random growth and random metric models in the Kardar–Parisi–Zhang (KPZ) universality class…

Let $\mathcal{L}$ be a directed landscape restricted to times in the interval $[0,1]$, and let $\mathcal{B}$ be a parabolic Airy line ensemble. Let $f$ be the function satisfying ……

The directed landscape $\mathcal L$ is the four-parameter scaling limit of Brownian last passage percolation: it encodes the last passage value from line $m$, location $x$ to line…

Canonical scaling-limit conjecture. The space-time Airy sheet should be the canonical scaling limit for last passage percolation models within the Kardar--Parisi--Zhang universalit…

Let $\Pi=\Pi_{0,0}$ be the directed geodesic from $(0,0)$, and define the increment process … For $0\le t<u<1/2$, absolute continuity conjecture. The laws of $\eta_t$ and $\eta_u$…

Let $\mathcal{A}$ be the Airy line ensemble, and define $\mathcal{H}:\mathbb{R}^2\to\mathbb{R}$ deterministically from $\mathcal{A}$ by the formulas in the source on…