The Maker-Breaker percolation game on the square lattice

We study the ( m , b ) Maker-Breaker percolation game on Z 2 , introduced by Day and Falgas-Ravry. In this game, on each of their turns, Maker and Breaker claim respectively m and b unclaimed edges of the square lattice Z 2 . Breaker wins if the component containing the origin becomes finite when his edges are deleted from Z 2 . Maker wins if she can indefinitely avoid a win of Breaker. We show that Breaker has a winning strategy for the ( m , b ) game whenever b ⩾ ( 2 − 1 / 14 + o ( 1 ) ) m , breaking the ratio 2 barrier proved by Day and Falgas-Ravry. Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the ( m , 2 m ) game even if he allows Maker to claim c edges before the game starts, for any integer c , and that he can moreover win rather quickly as a function of c . We also consider the game played on the so-called polluted board, obtained after performing Bernoulli bond percolation on Z 2 with parameter p . We show that for the ( 1 , 1 ) game on the polluted board, Breaker almost surely has a winning strategy whenever p ⩽ 0.6298 .