Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 3

The van den Berg-Kesten inequality says that for increasing cylinder events $A$ and $B$ under product bond percolation,where $A\square B$ is the event that $A$ and $B$ have disjoint finite witness edge sets.

Let $A_n$ be the increasing cylinder event that an open path joins $0$ to the boundary of the box $[-n,n{]}^d$. If two edge-disjoint open paths run from $0$ to infinity, then $A_n\square A_n$ occurs for every $n$. Therefore the van den Berg-Kesten inequality giveswhere continuity from above identifies ${\lim }_n{\mathbb{P}}_p(A_n)=\theta (p)$.

For $\omega \in \{0,1{\}}^{E(G)}$, write $o(\omega )$ and $c(\omega )$ for its numbers of open and closed edges and $k(\omega )$ for the number of connected components of the open spanning subgraph. The free random-cluster model is

Put $s=1-p$. On the four-cycle, the total unnormalized weight isThe event $N\leftrightarrow S$ contains all configurations with zero or one closed edge and exactly two of the six configurations with two closed edges, so its weight isDivide numerator and denominator by $q$. As $s\to 0$ and $sq\to \infty$, the omitted numerator terms are $o(1+2s^{2}q)$, while the middle denominator terms areConsequently

Take $A=\{N\leftrightarrow S\}$ and let $s=1-p\downarrow 0$ withThen $sq\to \infty$, $s^{2}q\to 0$, and $s^4{q}^3=s^{-1/2}\to \infty$. Part d gives ${\mathbb{P}}_{\mathrm{FK}}^{p,q}(A)\to 0$.

On a four-cycle, $A\square A$ occurs exactly when all four edges are open, because the two length-two paths from $N$ to $S$ are the only disjoint witnesses. HenceChoosing $s$ sufficiently small gives the two numerical inequalities in the question.