30 problems

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…

Equality of critical values conjecture.

Liggett–Wald conjecture. If $\lambda\leq\lambda_2(T)$, then

Consider the spatial contact process with mutations on a graph, with birth rate $lambda$ and mutation probability $r$, where pathogens of the same type die simultaneously and each…

Let $T_d$ be the homogeneous tree in which every vertex has $d+1$ neighbors, and consider the spatial contact process with mutations on $T_d$, where $\rho$ denotes the mutation pro…

Let Model S2 and Model S3 be the spatial pathogen models on ${\bf Z}^d$, with birth-rate parameter $\lambda$ and mutation probability $r\in[0,1]$. Write $r_c$ for a proposed critic…

Consider the multitype contact process with interaction range $r$ in dimension $d$, infection parameters $lambda1$ and $lambda2$, and frozen-state parameter $gamma$. Let $r0$ an…

Let $\mu$ be a probability distribution on the non-negative integers satisfying … Let $(\mathscr{G}_n)_{n\in\mathbb{N}}$ be configuration graphs on $n$ vertices derived from $\mu$,…

Subcritical phase conjecture. Then $\lambda_\mathsf{c}(\mathscr{G})=\lambda_\mathsf{c}>0$ $\mathbf{P}$-almost surely, and the contact process on $\mathscr{G}$ has a subcritical pha…

Can's conjecture. Can conjectured that $\delta^\ast\leq 2$; the cited result establishes only the bound $\delta^\ast\leq 102$.

Let $[?]$ be the payoff parameters in the multitype contact process with density-dependent birth rates, where $a_{11}$ and $a_{12}$ describe effects involving strategy 1 and…

Consider the one-dimensional contact process at its critical infection parameter, and modify the infection rate on a finite number of edges, allowing those rates to be arbitrarily…

Consider the long-range contact process (LoRaC) on the spatial model of the paper, with edge length $|y-x|$ and a parameter $\alpha>0$. Fewer edges. The LoRaC has a phase transitio…

Let $q^{(t)},p\in(0,1)$ be given and let $\alpha>1$. Let $q^{(x)}$ denote the parameter controlling the sparse spatial environment in the long-range contact process (LoRaC). Extinc…

Let $q^{(t)},q^{(x)},p\in(0,1)$ be given, and let $\alpha>1$. The long-range contact process (LoRaC) is the percolation model described in the paper, with edges whose length distri…

Positive critical-rate conjecture. For every $\gamma>0$ and $q\in[0,1]$,

Fast-update limit conjecture.

Traveling-wave uniqueness and velocity conjecture. There exist unique traveling waves, with wave velocity proportional to the width of $J$. The source presents this as a more speci…

Stable traveling-wave conjecture. For $k>1$, stable traveling-wave solutions exist, and they exhibit qualitative differences from the case $k=1$. Traveling fronts are known in the…

Let $J$ be the interaction kernel and let $\bar{\mathbf v}$ denote the sustaining stationary state. Consider initial data equal to $\bar{\mathbf v}$ on the region … and equal to th…

Let $bb_c(bc)$ denote the survival critical value of the symbiotic contact process with parameter $bc$. Critical-value separation conjecture. If $bc<1$, then … This would establish…

Let $\mu$ be a degree distribution, let $\mu'$ be its size-biased distribution, and write $\operatorname{GW}(\mu')$ for the Galton–Watson tree with offspring distribution $\mu'$. L…

Let $\xi$ be an offspring distribution satisfying … for all sufficiently large $k$, and consider the contact process on the Galton–Watson tree with offspring distribution $\xi$. Th…

Critical branching-rate conjecture. In the general model,

Let $(G_n)$ be a sequence of finite graphs converging locally to a limiting graph $G$, and let $\tau_n$ denote the extinction time of the contact process on $G_n$. Local-limit crit…