27 problems

Let $X=\mathrm{O}^-(6,q)$ be one of the $3$-transitive orthogonal polar graphs, where $q$ is a prime power and $q>3$. A graph has quantum symmetry if its quantum automorphism group…

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 Bernoulli percolation on a transitive graph in a parameter regime where there are almost surely infinitely many infinite clusters. Two infinite clusters are said to have i…

Benjamini–Lyons–Schramm conjecture. For every $p\in[0,1]$,

Let $G$ be a vertex-transitive graph with superlinear growth, and let $t_{\mathrm{c}}(\lambda)$ denote the critical lifespan for particle density $\lambda$. Positive critical lifes…

Let $G$ be a transitive graph with non-negative Ollivier--Ricci curvature. Recall that two metric spaces are quasi-isometric if they are equivalent up to multiplicative and additiv…

Maximal-displacement conjecture. The maximal displacement for a supercritical branching random walk on a transitive graph should exhibit the same behaviour; namely, if $\alpha$ is…

For a positive integer $n$, let the order density of a class of graphs mean the proportion of positive integers up to $n$ that occur as orders of graphs in that class. Order-densit…

Let $G$ be a transitive graph that is not one-dimensional, let $x$ be a vertex of $G$, and let $\deg(x)$ denote its degree in the random open subgraph. Uniform expected-degree conj…

Let $H$ be an infinite transitive graph, let $p_c(H)$ be its critical percolation threshold, and let $p_u(H)$ be its uniqueness threshold, the infimum of parameters for which there…

Easo–Hutchcroft conjecture. There exists a universal constant $C$ such that

Let $G$ be an infinite, connected, transitive, simple graph of vertex degree $d$ that is not one-dimensional, and let $p_c(G)$ be its Bernoulli bond percolation critical probabilit…

Let $\mathcal G_d^*$ be the class of infinite transitive graphs of degree $d$ that are not one-dimensional. For a graph $G$ with origin $o$, let $S_r$ be the sphere of radius $r$ a…

Let $\mathcal G_d^*$ be the class of infinite transitive graphs of degree $d$ that are not one-dimensional. For a graph $G$ with origin $o$, let $S_r$ be the sphere of radius $r$ a…

Benjamini's conjecture. The giant cluster should always be unique in the supercritical regime, irrespective of the geometry of $G$.

Let $G=(V,E)$ be a locally finite, connected, transitive, transient graph with spectral radius $\rho$, fix $o\in V$, and let $(X_n)_{n\geq 0}$ be simple random walk started at $o$.…

The first-return asymptotic conjecture. One has $f_n\asymp u_n$, and more precisely

Consider simple random walk $(X_n)_{n\geq 0}$ on a locally finite, connected, infinite, transitive graph $G$, with $n$-step return probability $u_n=\mathbb{P}[X_n=X_0]$ and spectra…

Tang's connectedness conjecture. For every $k\ge 3$ and every finite vertex-transitive graph $H$, the $\mathsf{FUSF}$ on $\mathbb{T}^k\times H$ is connected almost surely.

Let $G$ be a transitive graph of exponential growth, let $o$ be a vertex of $G$, and let $K_o$ be the cluster of $o$ in critical Bernoulli bond percolation. Write $\mathbb{P}$ for…

Let $\Gamma$ be a two-ended transitive graph. A finite Hamilton cover is a finite collection of Hamiltonian spanning subgraphs whose union covers the relevant graph structure, as i…

Consider Bernoulli bond percolation on a transitive graph, and let its percolation threshold be the critical probability separating almost-sure absence of an infinite cluster from…

Let $G$ be a general transitive graph, and consider independent percolation on $G$. A graph is non-amenable when its isoperimetric constant is positive, and a non-uniqueness phase…

Let $G$ be a transitive graph, let $B_G(u,n)$ be the radius-$n$ ball about $u$, and let $\partial B_G(u,n)$ be its boundary. For simple random walk on $G$, write $\mu_n(x)$ for the…