12 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 $\varepsilon$,

Let $S$ be a simple random walk on $\mathbb{Z}^{4}$ started at the origin. For $0\leq m\leq n$, let ${\cal G}_{m,n}$ be the random graph formed by the sites visited by $S$ from tim…

Let $S$ be a simple random walk on $\mathbb{Z}^{d}$ started at the origin. For $0\tsymm m\leq n$, let ${\cal G}_{m,n}$ be the random graph whose vertices are the sites visited by…

Let $G$ be a graph with edge weights, let $\lambda_2(G;\omega)$ denote its algebraic connectivity, and let $R(G;\omega)$ denote its total effective resistance. Define … A weight di…

Benjamini–Kozma conjecture. The quantity

The effective-resistance susceptibility conjecture. There exists a non-decreasing, diverging function $f:\mathbb{R}_+\to\mathbb{R}_+$ such that

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…

Bounded-degree small-influence central limit conjecture. Bounded degree and small influences should be sufficient to obtain a central limit theorem for the effective resistance.

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…