8 problems

Higher-dimensional max-flow min-cut conjecture. There exists a map $\mu$ on the unit sphere such that, for every convex set $A$ with $0$ in its interior,

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 $\varepsilon$,

Recursive limit-law conjecture. The limit law of $F$ is determined recursively as follows. If $P$ is a normal pattern, then $F$ is a Poisson pattern; in all other cases for $P$,…

Let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ be a biased disordered random network, with $d\geq 3$ and $0<\lambda_1\leq 1<\lambda_2$. A network is…

Let $p_c$ denote the critical probability for Bernoulli bond percolation on ${\mathbb{Z}}^d$, and let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ den…

Multilayer random-network conjecture. Under these conditions,