Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 214 / 2

Write $N_m=|B(m)|$ and $\theta =\theta (p)>0$. By translation invariance and linearity of expectation,Let $r={\mathbb{P}}_p(R(m)\ge \theta N_m/2)$. Since $0\le R(m)\le N_m$, splitting the expectation at the threshold givesRearranging produces the slightly stronger estimateOnly the mean and boundedness of the random variable are used; independence of the vertex-membership events is unnecessary.

For $m\ge 1$, let $S_m$ be the set of vertices of $B(m)$ connected to $\partial B(m)$ by an open path in a graph using only edges of $B(m)$. Every infinite-cluster vertex of $B(m)$ belongs to $S_m$, by stopping an infinite open path in a graph at its first boundary visit. Hence the increasing eventhas probability at least $\theta (p)/2$ by part (a). Importantly, $E_m$ depends only on the edges with both endpoints in $B(m)$.

Open all $8m$ perimeter edges joining successive vertices of $\partial B(m)$. They form a cycle in a graph, and opening them joins every vertex of $S_m$ into one percolation cluster, containing the specified perimeter vertex $v_m=(m,0)$. The Harris-FKG inequality bounds the probability of this event together with $E_m$ below by $p^{8m}\theta (p)/2$.

Close all $8m+4$ edges in the edge boundary of $B(m)$. These edges are distinct from the internal edges already considered, so their states are independent of those events. The resulting percolation cluster of $v_m$ is finite, contained in $B(m)$, and has at least $|S_m|$ vertices. Therefore, by translating $v_m$ to the origin,For $n\ge 1$, choose $m=\lceil \sqrt{n/(2\theta (p))}\rceil$. Then the threshold is at least $n$, and $m\le (1+(2\theta (p){)}^{-1/2})\sqrt{n}$. Put $A=-\log p$ and $B=-\log (1-p)$; both are positive. The preceding lower bound is at least $e^{-c_1\sqrt{n}}$ with, for example,The construction modifies only order-$m$ boundary edges while trapping order-$m^2$ vertices, which explains the stretched exponential scale. Here $\mathbb{N}$ is interpreted as the positive integers; the proposed lower bound at $n=0$ would be false because $\theta (p)>0$.

Declare an edge of the planar dual graph open exactly when its crossed primal edge is closed. This dual bond percolation has parameter $q=1-p<1/2=p_c$, using the Harris-Kesten theorem. We use the standard exponential tail of subcritical cluster size: for some $K,a>0$, uniformly in the dual vertex $v$,This is decay of the number of vertices in the percolation cluster, not merely its radius.

On $\{|C|=n\}$, the geometric fact allowed in the paper supplies a simple dual cycle in a graph surrounding $C$, with length $\ell \ge \alpha \sqrt{n}$. Every crossed edge is in the edge boundary of $C$ and is therefore closed, so the dual cycle is open. Its $\ell$ distinct vertices lie in one dual percolation cluster of size at least $\ell$.

A dual cycle in a graph of length $\ell$ surrounding the origin has all its vertices within sup-norm distance $\ell +1$ of the origin: its coordinate spans are at most $\ell$, and the origin lies between each pair of extreme coordinates. There are at most $K_0(\ell +1{)}^2$ possible dual vertices there. The union bound over lengths and possible vertices yieldsfor some $K_1,b>0$, since a polynomial factor can be absorbed into a slower exponential decay.

To remove the constant prefactor for all $n\ge 1$, first choose $n_0$ so that the bound is at most $e^{-(b/2)\sqrt{n}}$ for $n\ge n_0$. For the remaining finitely many $n$, use ${\mathbb{P}}_p(|C|=n)\le 1-\theta (p)<1$ and chooseFor $1\le n<n_0$, the chosen $c_2$ ensures $e^{-c_2\sqrt{n}}\ge 1-\theta (p)$.