The game behind oriented percolation

Ziliotto Bruno, CNRS, CEREMADE, Paris Dauphine University

We consider two opposing players who move a token along the edges of $\mathbb{Z}^2$, taking turns, one in the vertical direction and the other in the horizontal direction. Player 1 must pay a random cost to Player 2 for each edge crossed by the token. This cost is distributed identically and independently according to a Bernoulli distribution with a certain parameter $p$. Before the game begins, players are informed of all the costs. We study the value of the game $v(p)$ where the payment is the lower limit of the average payments. First, we show that this value is a constant. We also demonstrate that there exist $p_1$, $p_2$ in $(0,1)$ such that if $p \leq p_1$, then $v(p)=0$, and if $p \geq p_2$, then $v=1$. Furthermore, $v$ is continuous at $p_2$. Finally, we show that $p_2$ is equal to the critical threshold of classical oriented percolation on $\mathbb{Z}^2$. This critical threshold is defined as the infimum over $p \in [0,1]$ such that the event 'there exists an infinite component of 1' has a positive probability.

Area: CS31 - Random Games (Xavier Venel)

Keywords: Percolation, Zero-Sum, Long-term behavior

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