33 problems

Let $T$ be an infinite tree, and consider the contact process on $T$ started from a single infection. Write $\lambda$ for the infection rate and let the survival probability be the…

Let $T$ be an infinite tree. Call a vertex $v$ essential if $T\setminus\{v\}$ has at least two infinite components. Say that $T$ has strongly exponential growth if there is some in…

Let $T$ be an infinite tree, and let $\rho$ be its root. For the contact process on $T$, define … … … and … where $\overline\mu$ is the upper invariant measure and $\delta_0$ is th…

Positivity conjecture. If $s>2$, then

Pastor-Satoras–Vespignani conjecture. One has $\lambda_c=0$ when $\beta\leq 3$ and $\lambda_c>0$ when $\beta>3$.

Consider the other configuration models discussed in the source and their Galton--Watson local limits. Let the nonzero critical value denote the positive transition threshold for t…

Immunisation conjecture. If $\alpha\geq 1$, then for every sufficiently small $\kappa>0$ there exists $\nu'=\nu'(\kappa,\eta)$ such that

Fast-speed comparison conjecture. Under these assumptions, $\lambda_1>0$ if and only if $\lambda_1^p>0$, and $\lambda_2>0$ if and only if $\lambda_2^p>0$.

Slow-speed conjecture. The same phase-transition criterion holds throughout the slow-speed regime $\tau\leq 0$.

Asymptotic critical-line conjecture. There is a critical line asymptotically as $\kappa \to \infty$.

Let $X_t$ be the cooperative contact process, with parameters $\alpha,\delta$, and let $\eta(\alpha,\delta)$ be the infimum of initial product-law densities $p$ for which the law o…

Let $I_n(t)$ denote the expected number of infected vertices at time $t$ for the contact process on the dynamical graph with initially all vertices infected. Write…

Broman's conjecture. For large $k$, $(\xi_t^k)$ should behave like the ordinary contact process $(\eta_t)$.

Survival and strong-survival conjecture. The equivalence between finite survival and strong survival, known for the ordinary contact process, should also hold for the CPS.

Exponential estimates conjecture. If

Asymptotic shape conjecture. There exists a bounded convex set $U\subset\mathds{R}^d$ such that, for every $\varepsilon>0$,

Critical extinction conjecture. For every finite nonempty $C$ and every admissible background configuration $B$, one has

Let $d$ be the degree of the underlying regular graph, and let $\bar{\lambda}(\mathsf{v})$ denote the critical infection parameter for the contact process with switching rate…

Conjecture on extinction beyond ${\mathbb{Z}}$. The extinction result stating that, for every $p<p_c(d)$ and every infection rate $\lambda>0$, sufficiently slow edge updates cause…

Let $cmu$ be a degree distribution satisfying the giant component condition, and let $ctilde{\mu}$ be its size-biased distribution. Write $clambda_c^-(\mu)$ and $clambda_c^+(\mu)$…

Let the threshold-$\theta$ contact process evolve on a random $(\theta+1)$-regular graph, where an infected vertex requires at least $\theta$ infected neighbors for infection to pe…

Single critical value conjecture. The lower and upper critical values coincide:

Consider a random graph whose degree distribution satisfies … where $b>1$, and let $\lambda_n$ denote the infection rate. Start the contact process from the all-1 configuration. Cr…

Let the authors' general conditions be the hypotheses under which rescaled ranges of a single occupancy particle model converge to the range of super-Brownian motion. Range-scaling…

Consider the spatial cooperator-defector system with critical values $\beta_{\mathbf{c}}^-(\beta,\beta_{\mathbf{d}})$ and $\beta_{\mathbf{c}}^+(\beta,\beta_{\mathbf{d}})$, where…