3 problems

For every integer $d\ge 2$, consider the nearest-neighbor contact process on $\mathbb{Z}^d$ with infection rate $\lambda$ and recovery rate $1$. Let $\lambda_1$ be the critical val…

Consider the directed random graph on $\mathbb{Z}^2$ in which each vertex independently chooses one of its four nearest-neighbour directions uniformly, places an arrow in that dire…

For every $\delta>0$ there exists $\varepsilon>0$ such that the following holds uniformly over every finite graph $G$, every percolation parameter $p\in[0,1]$, every origin vertex…