The game of Hex, Bernoulli percolation, and probabilistic models of phase transition

Matan  Harel (Boston, MA)

Date: March 27, 2026

Abstract: The game of Hex is a two-player game played on an 11 x 11 rhombus, in which player one tries to create a blue left-right crossing, while player two seeks a red up-down crossing. What happens if we play this game randomly? What if we give one player an advantage? This simple setup of random geometry is a particular example of Bernoulli percolation, which is perhaps the simplest model of random geometry where we can observe a phase transition. More specifically, a very small change in a parameter (here, the probability of coloring a vertex blue) brings about a dramatic change of large-scale behavior. We will prove the existence of the phase transition and explore some properties that emerge from it.