11 problems

Strict inequality conjecture. For every nonamenable transitive graph $G$,

Let $G$ be a nonamenable transitive unimodular graph, and let $u{pc}(5mu)$ denote the speed of the random walk on critical dynamical percolation on $G$ with refresh rate $5mu$. Su…

Let $G$ be a nonamenable transitive graph, and consider critical Bernoulli percolation on $G$. Schonmann's conjecture. Critical percolation on every nonamenable transitive graph ex…

Bernoulli percolation uniqueness conjecture. Bernoulli percolation with $p$ sufficiently close to $1$ should have a unique infinite cluster, so that

Recall that the anchored Cheeger constant of a connected, locally finite graph $G$ is … where $v$ is a fixed vertex, and that $G$ has anchored expansion when $\Phi^*(G)>0$. Benjami…

Exponential connectivity decay conjecture.

$L^2$ boundedness conjecture.

Triangle-condition conjecture. For every $v\in V$,

The $L^2$ boundedness conjecture. If $G$ is nonamenable, then

We consider a hyperbolic lattice and a disk of large radius $R$ centered at an origin vertex. Boundary sites are vertices outside the disk connected to at least one vertex inside i…

Nonuniqueness conjecture. For every nonamenable transitive graph, there is some value of $p$ for which there is more than one infinite cluster almost surely.