The Maker-Breaker percolation game on the square lattice

TL;DR

This paper analyzes the Maker-Breaker percolation game on the square lattice, establishing Breaker's winning strategies under various conditions and extending previous results, especially around the ratio of claimed edges and percolation parameters.

Contribution

It proves Breaker can win when the ratio of edges claimed exceeds 2, and extends winning strategies to scenarios with initial edge claims and percolation effects.

Findings

Breaker wins for b ≥ (2 - 1/14 + o(1))m ratio

Breaker can win even with initial edge claims c

Breaker almost surely wins after percolation with p near 1/2

Abstract

We study the $(m,b)$ Maker-Breaker percolation game on $\mathbb{Z}^2$, introduced by Day and Falgas-Ravry. As our first result, we show that Breaker has a winning strategy for the $(m,b)$-game whenever $b \geq (2-\frac{1}{14} + o(1))m$, 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 $(m,2m)$-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 fast (as a function of $c$). Finally, we consider the game played on $\mathbb{Z}^2$ after the usual bond percolation process with parameter $p$ was performed. We show that when $p$ is not too much larger than $1/2$, Breaker almost surely has a winning strategy for the $(1,1)$-game, even if Maker is allowed to choose the origin after the board is determined.