What is it about?

Plant a forest with one thin tree at each vertex of \(\mathbb{Z}^d\). Then pick a tree "uniformly at random" and replace it by a lamp. Which are the trees lit by the lamp? We give a precise meaning to this question and answer it, by using both probability and number theory. We also solve generalisations of this problem. Using some results of Vardi, we show that, whenever \(d\geq2\), keeping the lit trees yields a unique infinite connected component (and finite components) almost surely while keeping the shady ones only yields finite components.

Electronic Communications in Probability, 27:1–14, 2022