The Maker-Breaker percolation game on a random board

The $(m,b)$ Maker-Breaker percolation game on (Z²)ₚ, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of Z² is removed independently with probability $1-p$. After that, Maker chooses a vertex v₀ to protect. Then, in each round Maker and Breaker claim respectively m and b unclaimed edges of G. Breaker wins if after the removal of the edges claimed by him the component of v₀ becomes finite, and Maker wins if she can indefinitely prevent Breaker from winning. We show that for any $p < 1$, Breaker almost surely has a wining strategy for the $(1,1)$ game on (Z²)ₚ. This fully answers a question of Day and Falgas-Ravry, who showed that for $p = 1$ Maker has a winning strategy for the $(1,1)$ game. Further, we show that in the $(2,1)$ game on (Z²)ₚ Maker almost surely has a winning strategy whenever $p > 0.9402$, while Breaker almost surely has a winning strategy whenever $p < 0.5278$. This shows that the threshold value of p above which Maker has a winning strategy for the $(2,1)$ game on Z² is non-trivial. In fact, we prove similar results in various settings, including other lattices and biases $(m,b)$. These results extend also to the most general case, which we introduce, where each edge is given to Maker with probability α and to Breaker with probability β before the game starts.