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

For each $x\in H_N$, the event that $x$ belongs to an infinite open cluster implies that $x$ has an open path to $h_N$. Translation invariance gives probability $\theta$ for the former event. Therefore

Write $M=\#H_N$, $X=\#C_N$, and $q=\mathbb{P}(X\ge \theta M/2)$. Since $0\le X\le M$,Thus $q\ge (\theta /2)/(1-\theta /2)\ge \theta /2$.

Both $\{\#C_N\ge \theta \#H_N/2\}$ and $\{0\in C_N\}$ are increasing events. The first has probability at least $\theta /2$ by part (a2), while the second has probability at least $\theta$ because an infinite origin cluster reaches $h_N$. The Harris-FKG inequality gives

On $A_N$, additionally require every site of $h_{N+1}$ to be open and every site of $h_{N+2}$ to be closed. The open ring joins every component of $C_N$ to the origin component, and the closed outer ring makes that component finite. It contains at least $\theta \#H_N/2$ sites. These ring states are independent of $A_N$, and their probability isSince both ring sizes are linear in $N$, this is at least $c(p)e^{-u(p)N}$. Multiplication by $\mathbb{P}(A_N)\ge {\theta }^2/2$ proves the claimed lower bound.

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.

A closed circuit of diameter at least $n$ through a fixed listed point contains a closed path from that point to graph distance at least $n/2$. Exponential decay of subcritical percolation bounds this probability by $C{e}^{-cn}$. A union bound over the $n$ listed points givesafter reducing $u>0$ and adjusting finitely many small $n$.

A closed circuit surrounding the origin and passing through $(m,0)$ has diameter at least $m$. By the one-point estimate used in part (b2), its probability is at most $C{e}^{-cm}$. Therefore

By planar duality on the triangular lattice, a finite open cluster containing the origin is surrounded by a closed circuit. If its diameter exceeds $n$, an outermost surrounding circuit reaches distance of order $n$ and crosses the positive coordinate ray at some $(m,0)$ with $m\ge cn$. Part (b3), with constants rescaled, therefore gives

A connected set of at least $m$ triangular-lattice sites has diameter at least $c_0\sqrt{m}$. Part (b4) givesafter reducing $v$ to absorb small $m$. Since $\#H_N$ is of order $N^2$, the lower bound in part (a4) is also of order $e^{-C\sqrt{m}}$. The upper and lower bounds therefore identify the correct stretched-exponential exponent $\sqrt{m}$.