Maker-Breaker Percolation Game on the Square Lattice

The (m,b) Maker-Breaker Percolation Game played on the edges of Z^2 has simple rules. Initially, all the edges of Z^2 are marked as unsafe. Maker and Breaker take turns. In each turn of hers, Maker marks m unsafe edges as safe. While in each of his turns, Breaker erases b unsafe edges from the graph. If the connected component of the origin ever becomes finite, Breaker wins, and else Maker wins.

Day and Falgas-Ravry proved that whenever b/m \geq 2, Breaker wins, and whenever b/m \leq \frac{1}{2}, Maker wins. In this talk, I will discuss an improvement for the side of Breaker. 

(Joint work with Adva Mond and Victor Souza.)