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

In site percolation on the three-dimensional cubic lattice, each vertex $v\in {\mathbb{Z}}^3$ is independently open with probability $p$ and closed with probability $1-p$. Thus ${\mathbb{P}}_p$ is the Bernoulli product measure on $\{0,1{\}}^{{\mathbb{Z}}^3}$, and open paths are nearest-neighbour paths all of whose vertices are open.

The percolation probability and percolation critical probability are

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.

For every $v\in {\mathbb{Z}}^3$, sample independent $U_v\sim \mathrm{Unif}[0,1]$ and set ${\eta }_p(v)=1_{\{U_v\le p\}}$. Then each ${\eta }_p$ has the required Bernoulli product law and ${\eta }_p\le {\eta }_{p^{\prime }}$ whenever $p\le p^{\prime }$. This is the monotone coupling of Bernoulli percolation.

Within the monotone coupling, $I_p$ increases with $p$ andTaking a countable cofinal sequence $p^{\prime }\uparrow p$ and using continuity from below of a measure givesSince $\{M<p\}\subseteq I_p$, subtraction from $\theta (p)=\mathbb{P}(I_p)$ proves

Use the standard theorem that supercritical Bernoulli percolation on ${\mathbb{Z}}^3$ has a unique infinite open cluster almost surely. Fix $p>p_c$ and choose $q\in (p_c,p)$. Almost surely ${\eta }_q$ has an infinite cluster somewhere. On $I_p$, the origin and that cluster lie in the unique infinite ${\eta }_p$-cluster, so a finite ${\eta }_p$-open path joins them. Almost surely every label on this finite path is strictly below $p$; increasing $q$ to some $r<p$ above those finitely many labels makes the origin percolate in ${\eta }_r$. Hence $I_p\subseteq \{M<p\}$ up to a null event, and part e gives left continuity at $p$. Together with part c, $\theta$ is continuous on $(p_c,1]$.