712 problems

Let $(G,o)$ and $(H,o')$ be rooted graphs, with the space of isomorphism classes of rooted graphs equipped with the local metric. Restrict to transitive graphs satisfying…

Let $G^+$ and $G^-$ be two copies of a finite graph $G$ with vertex labels $[n]:=\{1,\ldots,n\}$. For $H\subseteq[n]$, form the bunkbed graph $G^\pm=(V^\pm,E^\pm)$ with…

Consider a square lattice at the critical percolation point $p_c=1/2$, and its medial lattice with wired boundary conditions on one side and free boundary conditions on the other.…

Self-dual critical-point conjecture. In these systems, the critical point is conjectured to be

Let $Q^d$ be the graph with vertex set $\{0,1\}^d$ in which two vertices are adjacent if they differ in exactly one coordinate, and write $n:=2^d$ for its order. Let $Q^d_p$ be the…

Let $d\geq 2$ and consider i.i.d. bond percolation on $\mathbb{Z}^d$ with parameter $p>p_c(d)$. Let $\mathcal{C}_\infty$ be the almost surely unique infinite open cluster, let…

Let $D$ be a simply connected domain in the complex plane, and let $a,b,c,x$ be boundary points in counterclockwise order. Suppose that $D$ is conformally equivalent to the equilat…

Let $Q^t$ be the binary hypercube and let $Q^t_p$ be the random subgraph obtained by retaining each edge independently with probability $p$. A giant component is a connected compon…

Let $t(n,k)$ denote the rectangle crossing time, and let $(k_n)_{n\ge1}$ be a sequence with $k_n\to\infty$. Being above the median of $t(n,k_n)$ means the corresponding indicator p…

Consider the contact process on a connected graph, and modify the graph only locally while preserving connectedness. Pemantle–Stacey conjecture. The critical parameter of the conta…

Let $Gamma(0)subseteqmathbb{E}$ be the graph of infection from the origin in first-passage percolation, and let $K(Gamma(0))$ denote its number of ends. **Newman's infinite-ends c…

Higher-dimensional phase-transition conjecture. Similar results hold in higher dimensions whenever

Critical torus cluster-size conjecture. The largest cluster with periodic boundary conditions should have the same scaling as the largest cluster with zero boundary conditions, nam…

Let $r,s>0$, let $x\in\partial B(0,r)$, define … and let $\mathcal C_r(0)$ be the set of vertices in $B(0,r)$ connected to the origin by open paths contained in $B(0,r)$. Small-bou…

For a sequence of graphs whose local limits determine their local structure, consider the critical percolation probability $p_c$ for transitive graphs, where $p_c$ is the infimum o…

Long-cycle conjecture. Let $\varepsilon>0$ be a constant, and let

Let $G$ be a graph equipped with a nonunimodular transitive automorphism group, and let $p_h$ and $p_u$ denote the Hausdorff and uniqueness thresholds, respectively. Suppose that t…

CLE scaling-limit conjecture. CLE with parameter $\kappa$ is conjectured to be the scaling limit of the percolation interfaces in the critical $Q$-Potts random cluster model.

Critical percolation absence conjecture. If $p_c<1$, then there are no infinite clusters at $p_c$.

Let $H$ be an infinite transitive graph, let $\theta^H(p)$ denote the percolation order parameter at parameter $p$, and say that $H$ is one-dimensional in the sense used for transi…

Consider the high-temperature Ising model and critical Bernoulli percolation as two planar lattice models. High-temperature Ising–percolation universality conjecture. The universal…

Let $\mathcal{H}$ be a hypergraph and let $T\subseteq V(\mathcal{H})$ be a set of posts. In the conditioned model $E_4^T(\widetilde{\mathcal{H}})$, precisely one of the two copies…

Connectivity threshold conjecture. For the $G^{(k)}(n,r)$ filtered graph,

Recall that the Fortuin–Kasteleyn (FK) random-cluster model has cluster weight $q=1$. More generally, let $q\in(0,4]$, and set … For $\kappa\in[4,8)$, let…

Let $d\geq 2$ and consider Bernoulli bond percolation on $\mathds{Z}^d$, with critical probability … An edge is open when its Bernoulli($p$) state is $1$, and an open cluster is a…