Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 2 / b1

The variables ${\omega }^{\prime }(x)=1-\omega (x)$ are independent Bernoulli variables with parameter $1-p<1/2$. They therefore form subcritical site percolation on the triangular lattice; their open clusters are precisely the closed-site clusters of the original configuration.