12 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…

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 $\lambda_2^p>0$.

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…

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 $cmu$ 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 $V(R)$ be the number of potential range-$R$ neighbors and let $p_c(R)$ be the critical bond-percolation probability. For dimensions $d=2$ or $3$, let $\theta_c=\theta_c(d)>0$ b…

Let $G=(V,E)$ and $G'=(V,E')$ be connected graphs with the same vertex set $V$, where … for some $x,y\in V$. Write $\lambda_c(G)$ for the critical infection rate of the contact pro…