4 problems

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

Constancy conjecture. The function $\mathit\phi$ above is constant for each $\beta>0$.

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…