The game behind oriented percolation

TL;DR

This paper links the critical threshold of oriented percolation on ^2 to the value of a specially defined zero-sum game, revealing a deterministic relationship and continuity at the critical point.

Contribution

It introduces a zero-sum game framework to characterize the critical parameter of oriented percolation, establishing its determinism and continuity properties.

Findings

Game value equals 1 above critical percolation threshold

Game value is continuous at the critical point

Game value approaches 0 as percolation probability approaches 0

Abstract

We characterize the critical parameter of oriented percolation on $\mathbb{Z}^2$ through the value of a zero-sum game. Specifically, we define a zero-sum game on a percolation configuration of $\mathbb{Z}^2$, where two players move a token along the non-oriented edges of $\mathbb{Z}^2$, collecting a cost of 1 for each edge that is open, and 0 otherwise. The total cost is given by the limit superior of the average cost. We demonstrate that the value of this game is deterministic and equals 1 if and only if the percolation parameter exceeds $p_c$, the critical exponent of oriented percolation. Additionally, we establish that the value of the game is continuous at $p_c$. Finally, we show that for $p$ close to 0, the value of the game is equal to 0.