Consider $\mathbb{Z}^2$ as a graph, where each node has four neighbours. 4 signals are emitted from $(0,0)$ in each of four directions (1 per direction) . A node that receives one signal (or more) at a timestep will re-emit it along the 4 edges to its four neighbours at the next time step. A node that did not receive a signal at the previous timestep will not emit a signal irrespective of whether it earlier received a signal. There is a $50\%$ chance that a signal will be lost when travelling along a single edge between two neighbouring nodes. A node that receives more then 1 signal acts the same as if it received only 1. The emitting of a signal in each of the 4 directions are independent events.
What is the probability that the signal will sometime arrive at $(10^5,10^5)$? Research: Simulations show: Yes. ~90%
What is the probability that the signal will sometime arrive at $(x,y)$ if a signal traveling along an edge dies with probability $0 ? What is the least p, for which the probability that N initial random live cells die out approaches 0, as N approaches infinity? Experiment shows p close to 0.2872. In $\mathbb{Z}^1$, $p_{min}=0.6445...$, how to calculate this? In $\mathbb{Z}^3$, $p_{min}=0.1775...$, how to calculate this?