The Maker-Breaker percolation game on a random board

Abstract

The $(m,b)$ Maker-Breaker percolation game on $(\mathbb{Z}^2)_p$, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of $\mathbb{Z}^2$ is removed independently with probability $1-p$. After that, Maker chooses a vertex $v_0$ 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 $v_0$ 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 $(\mathbb{Z}^2)_p$. 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 $(\mathbb{Z}^2)_p$ 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 $\mathbb{Z}^2$ 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.

Figures (6)

• Figure 1: The known regions of the phase diagram for the $(1,1)$ game on $(\mathbb{Z}^2)_{\alpha,\beta}$. Almost surely, Breaker wins for points coloured red and Maker wins for points coloured blue.
• Figure 2: The known regions of the phase diagram for the $(2,1)$ game on the left, and for the $(1,2)$ game on the right, both played on $(\mathbb{Z}^2)_{\alpha,\beta}$.
• Figure 3: Clumps of size $1$ and $2$ produced by \ref{['alg:core-construction']} on a configuration with $p = 0.8$. Vertices are numbered as they appear.
• Figure 4: Disjoint triples of edges in $L_{v_0}^N$ used in Maker's strategy.
• Figure 5: Example of fully claimed $L_{v_0}^N$ in the protective game. Dual paths claimed by Breaker shown in red.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: Harris' inequality
• Proposition 3.1