Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 2 / c / Solution

The events $A_n$ decrease, and an open path from $0$ reaches every $\partial {\Lambda }_n$ exactly when the open cluster of $0$ is infinite. Therefore continuity from above of a measure givesEach $A_n$ depends on finitely many sites, so $p\mapsto {\mathbb{P}}_p(A_n)$ is a polynomial and hence continuous. Given $\epsilon >0$, choose $n$ with ${\mathbb{P}}_p(A_n)<\theta (p)+\epsilon$. For $p^{\prime }\downarrow p$, monotonicity and the finite-event continuity yieldThus $\theta (p^{\prime })\to \theta (p)$ from the right.