24 problems

Small-voter-fraction nonergodicity conjecture. For every $p<1/2$, there exists a number $\beta_0(p)>0$ such that, whenever $\beta<\beta_0(p)$, the hybrid process $\tilde\eta_t$ wit…

Let $(G_n)_{n\in\mathbb N}$ be a sequence of connected graphs, with $n$ vertices, edge set $E_n$, and maximum degree $\Delta_n$. Let $t_n$ be a non-negative sequence of times, let…

Dynamical exclusion-sensitivity threshold conjecture. If

Let $G$ be a realization of a directed configuration model with in-degree and out-degree sequences, and let $\chi$ denote the effective diffusion parameter of the voter model. Defi…

Consider a voter model on a possibly disconnected random graph, and define the consensus time as the first time at which every connected component reaches local consensus. Let the…

Let $G$ be a directed graph sampled from a directed configuration model (DCM) with degree sequences $(\mathbf{d}^+,\mathbf{d}^-)$, and let $\tau^u_{\rm cons}$ denote the consensus…

Let $T>0$, let $\nu_p$ be the product Bernoulli measure of density $p$, and let $\hat{\mathcal{Y}}^N$ and $\hat{\mathcal{Y}}$ be the discrete and limiting random fields defined in…

Two-dimensional occupation-time central limit conjecture. If $d=2$ and $\eta_0$ is distributed according to $\nu_p$, then

Consider the evolving scale-free network and the voter model in the same setup as Theorem, with arbitrary update rate $\kappa$. Let $T_{\text{cons}}$ denote the consensus time and…

Let $G_\kappa(t)$ be the evolving scale-free network with vertex updates, and consider the alternative dynamical-percolation model in which edges are updated independently at rate…

Persistence conjecture. When $q<1$, the process persists for time

Consider the biased voter model whose infection kernel is $a$, and let the left and right boundaries denote the corresponding interface boundaries. A kernel has a finite…

Cluster-size exponent conjecture. There exist exponents $\beta,\beta',\gamma,\gamma'$ such that

Crossing and displacement exponent conjecture. There exist $\nu,\delta>0$ such that

Let $B_L$ be the rhombic region of scale $L$, and let $Z_L$ denote the voter-model interface curve in $B_L$. Straight-line convergence conjecture. As $L\rightarrow\infty$, the inte…

Interface-length exponent conjecture. There exists $d\in[1,2]$ such that

Consider the one-dimensional constrained voter model with $F$ opinions and threshold $\theta$ on an infinite system. The system fluctuation-and-clustering conjecture. fluctuates an…

Let $\alpha=(\alpha_0,\alpha_1)$ be in the coexistence region of Theorem 1.10 of CDP11, with each $\alpha_i<1$ and sufficiently close to $1$. Let $\nu_{1/2}$ denote the nontrivial…

Let $p_t(x,y)$ be the transition probability at time $t$ for the rate-one random walk with kernel $p(,)$, and let $\blambda_p(kappa)$ denote the associated Lyapunov exponent. Suppo…

Let $\xi$ be the exclusion-voter mixture process with parameters $\beta,p$ and state space $\mathcal{D}$. A process is recurrent from $S_0$ when…

The passage-time moment conjecture. Suppose that $p\leq1/2$ and $\beta\in[0,1]$. For any $\varepsilon>0$ and $S_0\in\mathcal{D}\setminus\{\mathcal{D}_0\}$,

Let $\xi$ be the exclusion-voter mixture process with parameters $\beta,p$, relaxation time … and state space $\mathcal{D}$, with distinguished state $\mathcal{D}_0$. The process i…

Large-diffusion asymptotic conjecture. The limit

Voter-model Lyapunov-exponent conjecture. On $[0,\infty)$, $\kappa\mapsto\lambda_p(\kappa)$ is strictly decreasing and convex, with