6 problems

Inverse-square variance conjecture.

Bounded-variance conjecture. As $v$ tends to infinity,

Asymptotic shape conjecture. There exists a nonempty compact subset $B_0$ of $\mathbb{R}^2$ such that, for every positive number $\varepsilon$,

Graph-conditioning resistance conjecture. The process of conditioning a graph reduces the average resistance between any two vertices on the graph.

Let $T_{M^d}$ be the $d$-dimensional toroidal grid with all side lengths equal to $M$, and let $R_{\rm ave}(T_{M^d})$ denote its average effective resistance. More generally, let…

Let $G$ be an infinite resistance network with origin $o$, and let $v_x$ be the representative of an element of the energy kernel normalized by $v_x(o)=0$. Write $R^F(o,x)$ for the…