36 problems

Let $T$ denote the percolation time in the $r$-neighbour bootstrap percolation model on $\mathbb{T}_n^d$, and let $t=t(n)$ be a time parameter. Range conjecture. Theorem holds for…

Let $Q_d$ be the $d$-dimensional hypercube, and let $m(Q_d;4)$ denote the minimum cardinality of a set that percolates under the $4$-neighbour bootstrap process on $Q_d$. Exact-siz…

Let $C(k,l,m)$ be the cuboid consisting of $klm$ unit cubes, and call a configuration a solution when it has the dismantling property used throughout the paper. Small-cuboid existe…

Let $[n]^3$ be the $n\times n\times n$ cube, and let $s_3(n)$ denote the minimum possible area covered by the three orthogonal projections onto the faces of a solution to the faces…

Let $G=(V,E)$ be the adjacency graph of a uniformly repetitive (or uniformly recurrent) rhombus tiling. Let $F$ be a percolation process, monotone and freezing, on configurations…

Let $\square_{i=1}^n K_k$ be the Cartesian product of $n$ copies of the complete graph $K_k$, and let $p_c(G,r)$ denote the critical probability for $r$-neighbor bootstrap percolat…

Let $\mathcal{G}$ be the collection of all connected graphs of order at least $3$ with at most two blocks. A graph is 2-bootstrap good if it contains at least one pair of vertices…

Let ${\mathcal U}$ be a critical two-dimensional update family. Its difficulty is … where a critical update family is balanced if there are no two opposite directions $v,-v\in S^1$…

Let $M_4^3(n)$ be the maximum running time of the $K_4^3$-bootstrap process over all initial $3$-uniform hypergraphs on $n$ vertices. The paper gives a quadratic lower bound for th…

Let $\mathcal U$ be a critical $d$-dimensional family, meaning that its stable directions satisfy the criticality condition, and let $L_c(\mathcal U,p)$ be the critical length at i…

Let $(a,p,\theta)$ be fixed with $a>1$, and let $T_n^2$ be the two-dimensional torus. In the Bradonjić–Saniee bootstrap percolation model on the random geometric graph $G_{n,r}^2$,…

Let $(a,p,\theta)$ be fixed with $a>1$, and let $\theta_{\mathrm{local}}(a,p)$ denote the threshold for the appearance of large local, symmetrically distributed infectious outbreak…

Let $(a,p,\theta)$ be fixed with $a>1$, and let $G_{n,r}^2$ be the random geometric graph in the Bradonjić–Saniee bootstrap percolation model, with initially infected set $A_0$, fi…

Let $\mathcal U$ be a finite family defining two-dimensional $\mathcal U$-Ising dynamics. A finite set $\mathcal T$ of stable directions determines a $(\mathcal T,L)$-droplet when…

Consider the $K_d$-completion dynamics on $[n]$, in which copies of $K_d$ missing one edge are iteratively completed, with initial graph $G_0$ having edges $i\leftrightarrow j$ whe…

Critical-family universality conjecture. As $p\rightarrow 0$, either

Let ${\mathcal U}$ be a critical update family with an infinite number of stable directions. Let $q$ denote the vacancy probability, let $\tau_0$ be the infection time at the origi…

For the $\mathcal{U}$-bootstrap percolation model, let $q_{\mathrm{c}}$ denote its critical probability and let $\tilde q_{\mathrm{c}}$ denote the supremum of the directional criti…

Let ${\mathcal U}$ be an $\alpha$-unrooted and unbalanced critical two-dimensional update family with difficulty $\alpha$. Let $q$ denote the vacancy density and let $\tau_0$ be th…

For integers $m,n_1,\dots,n_m$ and parameters $\nu_{1}^{(1)},\dots,\nu_{n_1}^{(1)},\nu_{1}^{(2)},\dots,\nu_{n_2}^{(2)},\dots,\nu_{n_m}^{(m)}$, consider the functions $g_{\xi}$ aris…

Noise sensitivity conjecture. For any $d greater than or equal to k 2$, the event $\mathcal{C}n$ is noise sensitive.

Let $r\geq 3$ be fixed, and let $Q_d$ be the graph with vertex set $\{0,1\}^d$ in which two vertices are adjacent when they differ in exactly one coordinate. For the $r$-neighbour…

Higher-dimensional universality conjecture. begin{enumerate} item If $\mathcal{U}$ is subcritical, then

Fix an integer $d\geqslant2$ and let $\mathcal{U}$ be a $d$-dimensional update family. For $u\in S^{d-1}$, define the half-space … and the stable set … Let $\mu$ be Lebesgue measur…

Density-of-the-closure conjecture. For every $p\in[0,1]$, there exists a constant $\delta(p)$ such that $D(n)$ converges in probability to $\delta(p)$ as $n\rightarrow\infty$.