Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 204 / 3 / d / Solution

Work on the common probability space from part (c). For a configuration of all the uniforms, the set $S=\{r\in [0,1]:I_r\text{ occurs}\}$ is an upper interval, possibly with or without its lower endpoint. It is nonempty because $I_1$ occurs, and the coupled onset parameter for origin percolation is $M=\inf S$. For $0<p\le 1$,Indeed, occurrence at some $r<p$ implies $M<p$. Conversely, if $M<p$, the definition of infimum yields occurrence at some $s<p$, and a rational $r$ between $s$ and $p$ also satisfies $I_r$. This also proves measurability of $M$ through its strict sublevel sets.

Choose a rational sequence increasing to $p$. By continuity from below of a measure and the monotone coupling of Bernoulli percolation,Moreover, $\{M<p\}\subseteq I_p\subseteq \{M\le p\}$. Hence $I_p$ is the disjoint union of $\{M<p\}$ and $I_p\cap \{M=p\}$, givingIt is important to retain the intersection with $I_p$: occurrence at the infimum is not automatic. At $p=0$ there is no left limit within the parameter domain, so the displayed left-limit assertion is for $p>0$.