12 problems

Long-range percolation scaling conjecture. If $\alpha\in(0,2)$, then the corresponding random walk scales to a symmetric $\alpha$-stable Lévy process in $\mathbb R^d$.

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

Critical-exponent conjecture. (A) If $s=2$, then the diameter's order of magnitude is $N^\delta$, where $\delta$ is a function of $\beta$. (B) If $1<s<2$, then the diameter is…

Non-concavity conjecture. There exists $\delta=\delta(k,d)>0$ such that $p\mapsto q_c(p)$ is not concave on $[0,\delta]$.

Let $d\geq 1$, and let $J:\mathbb{Z}^d\times\mathbb{Z}^d\to[0,\infty)$ be a symmetric, integrable, translation-invariant kernel. Suppose that there are constants $0<\alpha<d$ and…

Consider the long-range percolation graph on $\mathbb T_n^d$ with all nearest-neighbour edges and each non-nearest-neighbour edge included independently with probability…

Consider the competing long-range infection model on $\mathbb T_n^d$ with long-range parameters $\alpha_\ominus(n)$ and $\alpha_\oplus(n)$, infection-rate ratio $\lambda(n)$, and c…

Let $d\ge2$ and let $p,q>d/2$ satisfy $1/p+1/q<2/d$. Consider symmetric long-range conductance configurations $\{c(x,y)=c(y,x):x,y\in\mathbb Z^d\}$ that are averaging and ergodic i…

Positive critical-rate conjecture. For every $\gamma>0$ and $q\in[0,1]$,

Fast-update limit conjecture.

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

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…