11 problems

Supercritical upper-bound conjecture. The displayed limsup is finite.

Finiteness conjecture. The corresponding directional limit should be finite (on the connected subsequence when nearest-neighbor bonds are not forced to exist).

The Boolean model is the weight-dependent random connection model with $gamma>0$, $gamma'=0$, and $delta=\infty$: each vertex has an independent Pareto-distributed radius with tail…

Let $d\geq 1$, let $J$ be a long-range percolation kernel with decay exponent $s>2d$, and let $D(\mathbf{0},x)$ denote the chemical distance between $\mathbf{0}$ and $x$. In the su…

Low-intensity Euclidean geometry conjecture. As $u\to 0$, the geometry of $\mathcal{I}^u$ converges to Euclidean geometry; equivalently, $D^u$, after the appropriate scaling, conve…

Let $\Gamma$ be a conformal loop ensemble with simple loops, ${\rm CLE}_\kappa$ for $\kappa \in (8/3,4)$, in a simply connected domain $D$, and let $\Upsilon$ be its carpet. For…

Small-intensity scaling conjecture. The iterated limit

Procaccia–Ye–Zhang's conjecture. The chemical distance in $\Gamma$ has the same order as the $\ell_{\infty}$-distance, in the sense established for supercritical Bernoulli percolat…

Fix $u>0$. Let $d_{\mathcal{FI}^{u,T}}(\cdot,\cdot)$ and $d_{\mathcal{I}^{u}}(\cdot,\cdot)$ denote the internal graph distances in finitary random interlacements (FRI) and random i…

Sharp covering-decay conjecture. There are an $\epsilon>0$ and a $\hat P_d\in(0,1)$ such that, for every $d\ge4$ and every such path satisfying $|\mathcal{P}|\le N^{1+\epsilon}$,

Let $G(X^{(n)})$ be the scale-free Gilbert graph on the point set $X^{(n)}$ in the torus $\mathbb{T}_n$, let $\mathsf{e}_1=(1,0,\ldots,0)$, and let $q(x)$ denote the closest point…