9 problems

Let $Z\subset\mathbb{R}$ be a nonempty finite set, let $g:Z\to\mathbb{R}$, let $Y\subset\mathbb{R}$ be a nonempty finite set, and let $f:Y\to\mathbb{R}$ satisfy … For $z\in Z$, def…

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 $a\to\infty$,

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…

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…

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 $X$ be the set of triples $(y,s,t)$ with $y\in\mathbb{R}$ and $s<t$. Equip $X$ with the metric … For $k\geq2$, let $E_k$ be the subset of triples with $|s-t|\geq1$ for which th…

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…