12 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…

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 $B_L$ 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…

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 $S_0\in\mathcal{D}\setminus\{\mathcal{D}_0\}$,

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